42
JHEP05(2014)064 Published for SISSA by Springer Received: February 12, 2014 Revised: April 4, 2014 Accepted: April 12, 2014 Published: May 15, 2014 Multiple scales in Pati-Salam unification models Florian Hartmann, Wolfgang Kilian and Karsten Schnitter Department Physik, Universit¨at Siegen, 57068 Siegen, Germany E-mail: [email protected], [email protected], [email protected] Abstract: We investigate models where gauge unification (GUT) proceeds in steps that include Pati-Salam symmetry. Beyond the Standard Model, we allow for a well-defined set of small representations of the GUT gauge group. We show that all possible chains of Pati- Salam symmetry breaking can be realized in accordance with gauge-coupling unification. We identify, in particular, models with unification near the Planck scale, with intermediate left-right or SU(4) quark-lepton symmetries that are relevant for flavor physics, with new colored particles at accessible energies, and with an enlarged electroweak Higgs sector. We look both at supersymmetric and non-supersymmetric scenarios. Keywords: Spontaneous Symmetry Breaking, GUT, Supersymmetric Effective Theories, Renormalization Group ArXiv ePrint: 1401.7891 Open Access,c The Authors. Article funded by SCOAP 3 . doi:10.1007/JHEP05(2014)064

JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

  • Upload
    lethuan

  • View
    224

  • Download
    0

Embed Size (px)

Citation preview

Page 1: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Published for SISSA by Springer

Received: February 12, 2014

Revised: April 4, 2014

Accepted: April 12, 2014

Published: May 15, 2014

Multiple scales in Pati-Salam unification models

Florian Hartmann, Wolfgang Kilian and Karsten Schnitter

Department Physik, Universitat Siegen,

57068 Siegen, Germany

E-mail: [email protected], [email protected],

[email protected]

Abstract: We investigate models where gauge unification (GUT) proceeds in steps that

include Pati-Salam symmetry. Beyond the Standard Model, we allow for a well-defined set

of small representations of the GUT gauge group. We show that all possible chains of Pati-

Salam symmetry breaking can be realized in accordance with gauge-coupling unification.

We identify, in particular, models with unification near the Planck scale, with intermediate

left-right or SU(4) quark-lepton symmetries that are relevant for flavor physics, with new

colored particles at accessible energies, and with an enlarged electroweak Higgs sector. We

look both at supersymmetric and non-supersymmetric scenarios.

Keywords: Spontaneous Symmetry Breaking, GUT, Supersymmetric Effective Theories,

Renormalization Group

ArXiv ePrint: 1401.7891

Open Access, c© The Authors.

Article funded by SCOAP3.doi:10.1007/JHEP05(2014)064

Page 2: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Contents

1 Introduction 2

2 Pati-Salam symmetry 3

3 General setup 4

4 Higgs mechanism and supersymmetry 9

4.1 Generic superpotential 9

4.2 Higgs mechanism and supersymmetry 10

5 Spectra and phenomenology 12

5.1 Mass matrix 12

5.2 Goldstone bosons 14

5.3 MSSM Higgs 15

5.4 The F multiplet 16

5.5 SM singlets 16

5.6 Matter couplings 16

5.7 Neutrino mass 17

6 Unification conditions 17

7 Supersymmetric Pati-Salam models 19

7.1 Extra mass parameters 20

7.2 General overview 20

7.3 Class E 22

7.4 Class F 25

7.5 Classes A to D 27

8 Pati-Salam models without supersymmetry 28

8.1 Class E 30

8.2 Class F 30

8.3 Class A to D 31

9 Summary of models 33

10 Conclusions 34

A Beta-function coefficients 35

B Vacuum expectation values and mass scales 36

C Model naming scheme 37

D Specific models 38

– 1 –

Page 3: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

1 Introduction

The Standard Model (SM) of particle physics organizes all known particles, including the

Higgs boson, in a linear representation of the space-time and gauge symmetries. This

representation is reducible and decomposes into several irreducible multiplets. There-

fore, the SM Lagrangian contains a considerable number of free parameters. A unified

theory (GUT) where the SM gauge symmetry is embedded in a larger gauge group can

reduce the number of free parameters. Furthermore, there is a general expectation that

a unified model of matter and interaction could explain some of the puzzles of particle

physics, such as quark and neutrino flavor mixing, the smallness of neutrino masses, or

dark matter.

Particularly promising GUT models are left-right symmetric extensions [1, 2], the Pati-

Salam model [3, 4], the SU(5) theory by Georgi and Glashow [5], or the combination of

both in an SO(10) [6] or E6 [7] gauge theory. In these GUTs, matter fields unify within

each generation, separately. The SM Higgs sector is enlarged. Further fields, symmetries,

or geometrical structure are required to explain the breaking of GUT symmetries and the

emergence of the SM (or its minimal supersymmetric extension, MSSM) at energies directly

accessible to us.

In this paper, we investigate various classes of Pati-Salam (PS) models. PS symme-

try, with a discrete left-right symmetry included, is the minimal symmetry group which

collects all matter fields of a single generation in one irreducible multiplet. As an exten-

sion of the SM, it is particularly interesting because it naturally accommodates multiple

energy thresholds or scales that fill the large gap between the SM (TeV) scale and the

scale of ultimate unification, presumably the Planck scale. Such models have been widely

discussed in case of both supersymmetric [8–23] and non-supersymmetric models [24–

26]. PS models do not just exhibit multiple scales associated with symmetry breaking,

but they can generate additional mass scales due to see-saw type patterns in the spec-

trum [27, 28].

We want to study the relation of symmetry breaking and intermediate mass scales

in PS models in a suitably generic way while keeping the sector of extra (Higgs) fields

below the GUT scale as compact as possible. To this end, we impose conditions on the

spectrum of new fields, so we can describe the chain of symmetry breaking via a staged

Higgs mechanism with only small representations of the GUT group. In this setup, we

scan over the complete set of models and investigate the allowed ranges and relations of

the various mass scales they can provide.

There are several specific questions that one can address by such a survey of models.

First of all, some GUT models are challenged by the non-observation of proton decay,

so we may ask whether complete unification, and thus the appearance of baryon-number

violating interactions, can be delayed up to the Planck scale. Secondly, whether the origin

of neutrino masses and flavor mixing could be associated with a distinct scale, decoupled

from complete unification. Thirdly, whether there can be traces of unification, such as

exotic particles, a low-lying left-right symmetry, or an enlarged Higgs sector, that may be

observable in collider experiments or indirectly affect precision flavor data.

– 2 –

Page 4: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

By restricting our framework to economical spectra with only small representations

(see section 3), we focus the study on phenomenologically viable models that look particu-

larly attractive. Nevertheless, they span a large range of intermediate-scale configurations

between the TeV and Planck scales. We do not impose extra technical conditions on the

gauge-coupling running and matching ([9, 12, 25]), so we can cover also corner cases where

scale relations are tightly constrained.

After a short review of PS models in section 2, we lay out the detailed framework of

our study in section 3. In section 4 we take a brief look at the staged Higgs mechanism,

before we discuss the overall spectrum patterns in section 5. In section 6 we implement the

conditions of gauge coupling unification. Finally, section 7 and section 8 present generic

properties and interesting details for the complete survey of PS models, divided in the two

classes of supersymmetric and non-supersymmetric models respectively, before we draw

conclusions.

2 Pati-Salam symmetry

By the label of Pati-Salam gauge group, we refer to the symmetry group

SU(4)C × SU(2)L × SU(2)R × ZLR . (2.1)

The SU(4)C factor is an extension of QCD with lepton number as fourth color, and there

is a right-handed SU(2)R symmetry analogous to the weak interactions. We impose an

exact discrete symmetry ZLR which exchanges SU(2)L with SU(2)R, and left-handed with

right-handed matter fields, respectively.

If we impose this symmetry on any generation of quarks and leptons, they combine

to a single irreducible representation ΨL/R. Regarding the continuous gauge group only,

this is a direct sum ΨL ⊕ ΨR, but it is rendered irreducible by the left-right symmetry.1

In particular, this representation necessarily includes a right-handed neutrino field. The

explicit embedding of the matter fields in the PS multiplet is as follows:

ΨL = (4,2,1) =

(ur ug ub ν

dr dg db e

)L

, ΨR = (4,1,2) =

(ucr u

cg u

cb ν

c

dcr dcg d

cb e

c

)R

. (2.2)

Here the subscript (r, g, b) are explicit color indices. This notation is compatible with the

usual conventions for supersymmetric GUT models, where we will denote by ΨL/R the

corresponding superfield.

The set of gauge bosons of a PS gauge theory contains the eight QCD gluons and the

weak WL triplet. On top of that, there are a WR triplet and seven extra gauge bosons

of SU(4)C . The latter consist of six (vector) leptoquarks and one SM singlet that gauges

the difference of baryon and lepton number, B − L. The SM hypercharge gauge boson

emerges as a linear combination of the two neutral PS gauge bosons. The discrete ZLR

1Throughout this paper, we indicate such Z2-irreducible direct sums by the subscript L/R, or simply

write Ψ for brevity.

– 3 –

Page 5: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

symmetry guarantees that the interactions of left-handed and right-handed fields coincide

for all representations.

The PS group is a subgroup of SO(10), but does not contain SU(5). It is well known

that the gauge couplings unify for a pure SU(5) GUT model, if supersymmetry is imposed.

However, the recent discoveries about neutrino masses and mixing (see review article in [29])

suggest additional new physics at high energies. Concrete estimates tend to place the

neutrino mass-generation scale below the GUT scale (either as direct mass scale [30] or

via a see-saw mechanism [31]). We should expect multiple thresholds where the pattern of

multiplets and symmetries changes. This idea fits well in the context of a PS symmetry

and its breaking down to the SM.

Therefore, we investigate scenarios where a PS gauge symmetry is valid above some

high energy scale. Below the PS scale, we allow for further thresholds where only a subgroup

of PS is realized. One such subgroup is the minimal left-right symmetry SU(3)C×SU(2)L×SU(2)R×U(1)B−L×ZLR. By construction, left-right symmetry breaking is associated with

flavor physics, both in the quark and lepton sectors.2 Above the PS scale, we maintain

the possibility of further unification to a GUT model. Examples are SO(10) [6] or a larger

group such as E6 [7].

PS models provide a partial explanation for the observed stability of the proton.

SU(4)C contains the U(1)B−L subgroup, so B − L is conserved. In analogy with QCD,

lepton number as the fourth color — and thus baryon number — is conserved by all inter-

actions that do not involve the totally antisymmetric tensor εabcd with four color indices.

Counting field dimensions, this excludes baryon-number violating interactions of fermions

in the fundamental or adjoint representations of SU(4)C , at the renormalizable level. In

a minimal PS extension of the SM, proton-decay operators are thus excluded or, at least,

naturally suppressed. In particular, the terms of the MSSM which mediate proton decay

via squark fields are inconsistent with PS symmetry.3

A field condensate that spontaneously breaks SU(4)C to SU(3)C carries lepton number,

but no baryon number. Hence, PS breaking does not induce proton decay either, at the level

of renormalizable SM or MSSM interactions. On the other hand, breaking PS symmetry

down to the SM gauge groups can generate and relate the operators that provide neutrino

Majorana masses and mixing in the lepton sector, as well as Yukawa terms and mixing in

the quark sector.

3 General setup

In this paper, we consider both supersymmetric and non-supersymmetric versions of a PS

gauge theory. We first investigate supersymmetric models, where the model space is more

constrained. For this purpose, we assume N = 1 supersymmetry to hold over the full

2In a left-right symmetric model, up- and down-type masses are degenerate, and all mixing angles can

be rotated away.3We note that supersymmetric PS models need not include R parity for eliminating proton-decay oper-

ators. However, for more general extensions of the MSSM we have to discuss this issue (see section 5.4).

For the sake of simplicity we imply R parity for all our models.

– 4 –

Page 6: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

energy range above the TeV scale. Later we will also discuss the running of the gauge

couplings in non-supersymmetric versions of the models.

Following a top-down approach, we put the highest scale under consideration at the

Planck scale (we use MPlanck = 1018.2 GeV). In that energy range, gravitation is strong

and a perturbative quantum field theory in four space-time dimensions is unlikely to be

an appropriate description. Below this scale, gravitation becomes weak, and an effective

weakly coupled four-dimensional gauge theory can emerge. This gauge symmetry may be

the PS group. We suspect that at the highest energies there is a larger local symmetry

(GUT) that contains PS as a subgroup. By MGUT we denote the energy scale where this

larger symmetry is broken down to PS. We note that MGUT may coincide with MPlanck,

and that the GUT theory need not be a perturbative field theory.

It is important to study possible mechanisms that can break a Planck-scale GUT sym-

metry down to the PS group. Unfortunately, few reliable and generic results are available.

For instance, the survival principle that restricts the pattern of multiplets in the low-

energy effective theory [32–35] does not hold in supersymmetric theories [27, 28, 36]. In

the vicinity of the Planck scale, all conceivable effects from string states, gravitation, extra

dimensions, strong interactions, and possibly completely unknown principles need to be

taken into account in the GUT theory. This is beyond the scope of this paper. Therefore,

we will leave unspecified the field content above MGUT and the mechanism responsible for

GUT breaking down to the PS group.

On the other hand, at energies significantly below the GUT scale, there are good

reasons to expect the effective theory as a (possibly supersymmetric) weakly interacting

four-dimensional quantum field theory, where further symmetry breakings are realized by

a conventional Higgs mechanism. We will take this as a working hypothesis. Without

solid knowledge about the available PS multiplets below the GUT scale, we could allow

for any set of group representations as the field content of the effective theory. Instead,

we follow a simple phenomenological approach and restrict our analysis to a set of small

representations. This set is chosen such that it is just sufficient to realize all possible chains

of further symmetry breaking.

Below MGUT, we are thus dealing with a PS model which we specify in more detail. In

the supersymmetric case, we list all the PS-symmetric supermultiplets that are contained

in the spectrum. All such supermultiplets can interact and contribute to the running of

the gauge couplings. The effective Lagrangian may contain all symmetric renormalizable

interactions of these superfields. It will also contain, in general, non-renormalizable in-

teractions of arbitrary dimension which are suppressed by powers of the relevant higher

mass scales, i.e., GUT or Planck scales. While non-renormalizable terms, as it turns out,

are not required for the symmetry-breaking chains that we discuss below, they can con-

tribute significantly to flavor physics in the low-energy effective theory. We will not discuss

non-renormalizable effects in this paper in any detail, so we do not attempt a complete

description of flavor. The necessary inclusion of all kinds of subleading terms is beyond

the scope of this paper.

For the supersymmetric PS model and the effective theories down to the SM, we state

the following simplifying assumptions:

– 5 –

Page 7: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Field(SU(4), SU(2)L,SU(2)R

)SO(10) E6

Σ (15,1,1) 45

78

TL ⊕ TR (1,3,1)⊕ (1,1,3)

E (6,2,2)

ΦL ⊕ ΦR (4,2,1)⊕ (4,1,2) 16

ΦL ⊕ ΦR (4,2,1)⊕ (4,1,2) 16

S78 (1,1,1) 1

ΨL ⊕ΨR (4,2,1)⊕ (4,1,2) 16

27h (1,2,2)

}10

F (6,1,1)

S27 (1,1,1) 1

Table 1. Chiral superfield multiplets of the PS models considered in this paper. The fields are

classified by their gauge-group quantum numbers; the discrete ZLR symmetry renders the multiplets

left-right symmetric and thus irreducible.

1. The model is effectively a perturbative quantum field theory in four dimension at all

scales below MGUT.

2. The breaking of the PS symmetry and its subgroups is due to Higgs mechanisms that

involve the superfields contained in the given PS spectrum.

3. SUSY is broken softly by terms in the TeV energy range.

4. The effective scale-dependent gauge couplings coincide at MGUT, consistent with a

GUT symmetry that contains SO(10).

5. The chiral supermultiplets all fit in the lowest-lying (1, 27, and 78) representations

of E6 or, equivalently, in 1, 10, 16, 16, and 45 representations of SO(10).

The last point is the most important restriction that we impose on our models. (For a recent

survey with different assumptions, see [9, 25]). As we will discuss below, this minimal field

content can realize all possible chains of PS sub-symmetry breaking and addresses the

phenomenological questions that we raised in the Introduction above. Simultaneously, a

model with only small representations can easily be matched to a more ambitious theory

of Planck-scale symmetry breaking. For instance, all multiplets that we consider, occur in

the fundamental 248 representation of E8 [37].

We now state explicitly the field spectrum that is consistent with our assumptions and

display all possible irreducible PS multiplets in table 1. The matter fields of the MSSM

are contained in ΨL/R multiplets. There are three copies of this representation, each one

including a right-handed neutrino superfield. The other supermultiplets are new. They

– 6 –

Page 8: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

contain the MSSM Higgs bidoublet, various new Higgs superfields, and extra “exotic”

matter. Let us take a closer look at these superfields:

1. h could directly qualify as the MSSM Higgs bi-doublet.

2. Σ and TL/R are chiral multiplets in the adjoint representation. They have the same

quantum numbers as the gauge fields of SU(4)C and SU(2)L,R, respectively.

3. ΦL/R and ΦL/R are extra multiplets with matter quantum numbers and their charge-

conjugated images, respectively. Both must coexist in equal number, otherwise ad-

ditional chiral matter generations would be present at the TeV scale.4

4. E and F contain only colored fields. Under SU(3)C they decompose into triplets

and anti-triplets, so they can be viewed as vector-like quarks and their scalar su-

perpartners. Depending on their couplings, they may also behave as leptoquarks or

diquarks. The F multiplet may have both leptoquark and diquark couplings and thus

can violate baryon number (cf. section 5.4). For the E multiplet, this is not possible.

Note that Σ also provides vector-like quark and antiquark superfields, together with a

color-octet and a color-singlet superfield. The Σ couplings conserve baryon number.

5. The singlet fields S78 and S27 (and any further singlet fields that originate from

SO(10) or E6 singlets) can couple to any gauge-invariant quadratic polynomial.

Chiral superfields which are not copies of the above are excluded by our basic assumptions.

An explicit (i.e., PS invariant) mass term in the superpotential can be either present

in the superpotential ab initio, or generated by a singlet condensate. Without further

dynamical assumptions, mass terms and mass thresholds are thus constrained only by

matching conditions on the coupling parameters, and may exhibit a hierarchical pattern.

Some of the above mentioned multiplets contain an electroweak singlet and thus qualify

as Higgs superfields for various steps of PS symmetry breaking, if the corresponding scalar

component acquires a vacuum expectation value (vev). The associated vevs may be denoted

as 〈T (1, 1, 3)〉, 〈Σ(15, 1, 1)〉, 〈Φ(4, 1, 2)〉, and 〈h(1, 2, 2)〉, corresponding to the Higgs fields

introduced below. They induce the following breaking patterns:

SU(4)C〈Σ〉−−→ SU(3)C ×U(1)B−L , (3.1a)

SU(2)R〈T 〉−−→ U(1)R , (3.1b)

SU(4)C ×U(1)R〈Φ〉−−→ SU(3)C ×U(1)Y , (3.1c)

SU(2)R ×U(1)B−L〈Φ〉−−→ U(1)Y , (3.1d)

U(1)R ×U(1)B−L〈Φ〉−−→ U(1)Y , (3.1e)

SU(2)L ×U(1)Y〈h〉−−→ U(1)E . (3.1f)

4A fourth chiral generation is not finally excluded, but unlikely in view of present data.

– 7 –

Page 9: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

SU(4)C ×SU(2)L×SU(2)R×ZLR

SU(3)C ×U(1)B−L×SU(2)L×SU(2)R×ZLR SU(4)C ×SU(2)L×U(1)R

SU(3)C ×SU(2)L×U(1)R×U(1)B−L

SU(3)C ×SU(2)L×U(1)Y

Figure 1. Graphical illustration of the different breaking paths depending on the different classes.

Class B is shown in red, C in green, and D in blue.

If there is a hierarchy between the vevs, we observe a cascade of intermediate symme-

tries. Thus, we have the possibility for multi-step symmetry breaking. In the following,

we distinguish six classes of PS models by their hierarchy patterns. We denote them as

follows:

A: 〈Φ〉 ∼ 〈T 〉 ∼ 〈Σ〉 (one scale),

B: 〈Φ〉 � 〈T 〉 � 〈Σ〉 (three scales),

C: 〈Φ〉 � 〈Σ〉 � 〈T 〉 (three scales),

D: 〈Φ〉 � 〈T 〉 ∼ 〈Σ〉 (two scales),

E: 〈Φ〉 � 〈Σ〉 and 〈T 〉 = 0 (two scales),

F: 〈Φ〉 � 〈T 〉 and 〈Σ〉 = 0 (two scales).

The symmetry breaking chains associated to these classes are shown in figure 1 (see

also [28]).

The scale 〈Φ〉 is always the lowest symmetry-breaking scale (above the electroweak

scale), since this vev breaks all symmetries. The other vevs break the PS symmetry only

partially and are only relevant if they lie above 〈Φ〉. Nevertheless they are needed because

we require the symmetry breaking to result from a renormalizable potential, which does not

exist for Φ alone. Class A and D are limiting cases of the other and will not be discussed

in greater detail.

The number of matter multiplets (ΨL/R) is fixed to three. All other multiplets may

appear in an arbitrary number of copies. We will specifically study scenarios where the

multiplicity is either zero, one, or three. This already provides a wide range of possibilities

for spectra and hierarchies and appears most natural in view of the generation pattern that

is established for SM matter.

– 8 –

Page 10: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Let us further categorize models, constructed along these lines, that we will study

below. Each model type may fit into any of the above classes. It contains the three

generations of MSSM matter superfields ΨL/R together with the following extra superfields:

Type m (minimal model). Within a given class, the minimal model is the one with

minimal field content that realizes the corresponding symmetry-breaking chain. In

classes A to D, this model contains a single copy of each of the multiplets Φ, Σ and

TL/R. In classes E (F), TL/R (Σ) is omitted, respectively.

Type s (SO(10)-like model). In this model type, the multiplets can be composed to

complete SO(10) representations. All multiplets listed in table 1 are present as a

single copy.

Type e (E6-like model). In this type of model, all fields of table 1 are present. The

multiplets h, F , and S27 appear in three copies since they combine with the matter

multiplets. For the other multiplets, we set the multiplicity to one.

Type g (generic model). Here we classify all models that do not qualify as either of the

above three types. We will denote these models by their class and a unique number.

The numbering scheme is described in appendix C.

4 Higgs mechanism and supersymmetry

Some of the superfields that we introduce beyond the MSSM matter fields, should act as

Higgs fields and break the gauge symmetry, step by step, down to the SM gauge group. For

each breaking step, we have to verify that the scalar potential has a local minimum for a

non vanishing field configuration that breaks the gauge symmetry in the appropriate way.

The value of the superpotential at this field configuration should be zero, so supersymmetry

is maintained in the ground state.

To achieve this, we construct a generic renormalizable superpotential for the multiplets

that we include in a specific model. Non-renormalizable terms could be added, but we will

see that the various routes of gauge symmetry breaking result from renormalizable terms

alone, so they can be ignored in a first approach. The specific models derive from a generic

superpotential that includes all allowed superfields and their renormalizable interactions.

4.1 Generic superpotential

We start at the GUT scale with the most general renormalizable PS-invariant superpoten-

tial for the superfields shown in table 1 and impose parity (ZL/R) as a symmetry at this

scale. The superpotential can be broken down in several parts which we organize in view

of the model types m, s, and e as introduced above.

W = WΦ/Σ/T +Wh/F +WE +WS +WYukawa (4.1)

The terms within WΦ/Σ/T generate the Higgs potential for all steps in the staged Higgs

mechanism. This superpotential consists only of those fields that are allowed to get a vev.

– 9 –

Page 11: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Depending on the desired vev structure (case A to F), we may set some terms to zero, to

obtain a minimal superpotential.

The superpotential Wh/F is present in the models that contain the fields h and F .

Similarly, the field E comes with the terms WE. The potential WS contains all interactions

of the singlets with the other Higgs fields, where in each term, S indicates an arbitrary

linear combination of all PS singlets present in the model.

If h, F are present, there is the WYukawa superpotential. This part, which is the only

renormalizable superpotential involving (two) matter superfields,5 implicitly contains gen-

eration indices. Analogously, generation indices are implied for all superfields that occur

in more than one copy. We note that Y contains only symmetric Yukawa couplings that

are universal across leptons, neutrinos, up- and down-type quarks.

WΦ/Σ/T = −mΦ

(ΦL ΦL + ΦR ΦR

)− 1

2mΣ Σ2 +

1

3lΣ Σ3 + lΣΦ

(ΦL Σ ΦL + ΦR Σ ΦR

)− 1

2mT

(T 2

L + T 2R

)+ lTΦ

(ΦL T L ΦL + ΦR TR ΦR

)(4.2a)

Wh/F = −1

2mh h

2 − 1

2mF F 2 + lhΦ h

(ΦL ΦR + ΦR ΦL

)+ lFΦ F

(ΦL ΦL + ΦR ΦR

)+ lF Φ F

(ΦL ΦL + ΦR ΦR

)+ lΣF F ΣF + lTh h

(T L + TR

)h (4.2b)

WE = −1

2mE E2 + lTE E

(T L + TR

)E

+ lΣE E ΣE + lFEh F E h (4.2c)

WS = −1

2mS S2 +

1

3lS S3

+ sΦ S(ΦL ΦL + ΦR ΦR

)+ sT S

(T 2

L + T 2R

)+ sΣ S Σ2

+ sh S h2 + sF S F 2 + sE SE2 (4.2d)

WYukawa = Y ΨL hΨR + YF F(ΨL ΨL + ΨRΨR

)(4.2e)

4.2 Higgs mechanism and supersymmetry

For our supersymmetric models, we want to keep SUSY unbroken down to the TeV (MSSM)

scale. To satisfy this constraint, we have to verify that all F and D terms vanish [38, 39]

after inserting the vacuum expectation values of the fields:6

Fi|vev = 0 and Da|vev = 0 . (4.3)

5Only ΨL and ΨR are considered as matter superfields, containing all fermions of the SM.6We note that these conditions are trivially satisfied if all expectation values vanish. Without specifying

further (possibly non-perturbative) dynamics, for a supersymmetric Higgs mechanism there is the alterna-

tive of no symmetry breaking, degenerate in energy. A complete theory should include a mechanism for

supersymmetry breaking that lifts this degeneracy.

– 10 –

Page 12: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

We explicitly calculate the scalar potential for the generic superpotential shown

in (4.1). This covers all models under consideration; for any specific model, we set the

couplings to zero that are not involved.

The fields that transform under PS but contain a singlet under the SM gauge group,

are ΦR, ΦR, Σ, TR, and the matter superfields ΨR. In particular, the SU(4)C singlets in

ΦR and ΦR simultaneously break SU(2)R and thus induce direct breaking PS → MSSM.

However, there is no nontrivial renormalizable superpotential of ΦR,ΦR alone, so we include

the Σ and TR multiplets in the discussion. The vevs 〈Σ〉 and 〈TR〉 are particularly relevant

to the symmetry-breaking chain if they are significantly larger than vΦ, since they leave

subgroups of PS intact (cf. figure 1). We introduce the following notation:

〈ΦR〉 = ΦR4,1,1 = vΦ , (4.4a)

〈ΦR〉 = ΦR4,1,1 = vΦ , (4.4b)

〈Σ〉 = Σ15,1,1 = vΣ , (4.4c)

〈TR〉 = T1,1,3 = vT . (4.4d)

Here the subscript denotes the field component (or generator) that acquires the vev. These

are all vevs we allow for in this paper.

For flavor physics, the vevs vΦ and vT are of particular interest, because they break

the discrete ZLR symmetry. As long as the left-right symmetry is unbroken, there are no

terms that distinguish right-handed up-type from down-type quarks. Therefore, the energy

scale below which flavor mixing appears in the renormalizable part of the effective theory,

is given by vΦ or vT . On the other hand, both vΣ and vΦ separate quarks from leptons, so

only in class-F models where vΣ = 0, we expect a direct relation between lepton-flavor and

quark-flavor mixing.

The ground-state values of D terms are zero if and only if the vevs of mutually conjugate

fields exist simultaneously and coincide in value. For T and Σ the D-terms automatically

vanish since their generators are traceless. Therefore, we must have 〈ΦR〉 = 〈ΦR〉:

vΦ ≡ vΦ . (4.5)

The vacuum expectation values of the F terms also have to vanish. Inserting the vevs,

we obtain the necessary and sufficient conditions

0 = FΦ4 = vΦ(mΦ − lΣΦ vΣ − lTΦ vT ) , (4.6a)

0 = FΣ15 = vΣ(mΣ − lΣ vΣ)− lΣΦ v2Φ , (4.6b)

0 = FTR3

= mT vT − lTΦv2Φ , (4.6c)

0 = FS = sΦ v2Φ + sΣ v

2Σ + sT v

2T . (4.6d)

– 11 –

Page 13: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Solving for the mass parameters of Φ, T and Σ as well as for one of the singlet couplings,

we obtain

mΦ = lΣΦ vΣ + lTΦ vT , (4.7a)

mT =lTΦ v

vT, (4.7b)

mΣ =lΣΦ v

vΣ+ lΣ vΣ , (4.7c)

sΦ = −sΣ v

2Σ + sT v

2T

v2Φ

. (4.7d)

We have verified that these vev configurations also minimize the scalar potential while

maintaining supersymmetry. Thus, depending on their mutual hierarchy, they realize the

symmetry-breaking chains of the model classes A to D.

For the models classes E and F, we assume a vanishing vev of T or Σ, respectively.

Nevertheless, PS symmetry is broken completely down to the MSSM gauge symmetry.

A vanishing vev results if the corresponding multiplet does not couple to ΦR/ΦR. This

can be realized by either omitting the multiplet entirely, or by setting lΣΦ (lTΦ) to zero,

respectively. Otherwise, SUSY would be broken.

As long as we stay in the regime of renormalizable superpotentials, it is not allowed

to set vΦ = 0, because in this case the solution of the minimization condition would be

vT ≡ 0, so SU(2)R would remain intact. The reason is the absence of trilinear terms for

TR in the renormalizable superpotential: SU(2) has no cubic invariant. In addition there

would be no breaking of U(1)R ×U(1)B−L down to hypercharge without vΦ.

5 Spectra and phenomenology

The various classes of models that we introduced above, lead to a great variety in the

observable spectra. In the current work, we aim at a qualitative understanding, so we

refrain from detailed numerical estimates or setting up benchmark models. Nevertheless,

we observe a few characteristic patterns which can have interesting consequences for collider

and flavor physics. We discuss these patterns and their phenomenological consequences

below.

5.1 Mass matrix

Having verified that SUSY remains unbroken down to the TeV scale where soft-breaking

terms appear, we consider the mass matrix of the scalars. We do not intend to derive

quantitative results here, but just assign mass scales to the individual multiplets which

depend on the hierarchy of the vevs. In table 2, we list the results for all fields and the

interesting model classes B, C, E and F. The classes A and D are limiting cases. Therefore

they are not listed separately.

We should keep in mind that this table refers only to the mass contributions that

result from symmetry breaking. All superfields except for the matter multiplets, may

carry an individual PS-symmetric bilinear superpotential term. These mass terms are

– 12 –

Page 14: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Field (SU(3)c, SU(2)L)Y B C E F

Σ (8,1)0 vΣ vΣ vΣ −

E (3/3,2)± 56

− − − −

E (3/3,2)± 16

− − − −

ΦL/ΦL (3/3,2)± 16

vΣ vT vΣ vT

ΦR/ΦR (3/3,1)± 23

vΣ vΣ vΣ vΦ

Σ (3/3,1)± 23

vΣ vΣ vΣ −

ΦR/ΦR (3/3,1)± 13

vΣ vT vΣ vT

F (3/3,1)± 13

v2Φ

v2Φ

vT

v2Φ

v2Φ

vT

TL (1,3)0v2Φ

vT

v2Φ

vT− v2

ΦvT

ΦL/ΦL (1,2)± 12

vT vT vΦ vT

h (1,2)± 12

v2Φ

vT

v2Φ

vTvΦ

v2Φ

vT

ΦR/ΦR (1,1)±1 vT vT vΦ vT

TR (1,1)±1 vT vT − vT

TR (1,1)0 vΦ vT − vΦ

Σ (1,1)0 vΣ vT vΣ −

ΦR/ΦR (1,1)0 vΦ vΣ vΦv2Φ

vT

S27/S78 (1,1)0v2Σ

vT

v3Σ

v2T− −

S27/S78 (1,1)0v2Σ

vT

v3Σ

v2T

vΦv2Φ

vT

Table 2. Mass hierarchy of the scalar fields in the different classes of the complete model. If

non is shown, there is no contribution from the vev and their hierarchy is undefined. Class B:

vΦ � vT � vΣ; class C: vΦ � vΣ � vT ; class E: vΦ � vΣ, vT = 0; class F: vΦ � vT , vΣ = 0; the

classes A and D are a limiting case of B and C. Class A can be reached if one sets all vevs equal to

a single vev v and class D if one just sets vΣ = vT = v. The order is thus, that fields which mix are

grouped together. Thus the mass eigenvalues are a linear combination of the listed fields. Massless

components which are the Goldstone Bosons are explicitly not considered here.

completely arbitrary and, with supersymmetry, do not cause naturalness problems [40, 41].

Furthermore, a vev in any PS singlet field may contribute a similar term, again unrestricted

by symmetries. Hence, by including a multiplet in the spectrum of the effective theory in a

particular energy range, we implicitly assume that the sum of all mass terms for this field

is either negligible compared to the energy, or fixed by the conditions that determine the

vacuum expectation values.

On the other hand, in many models, not all of the fields E, Σ and T receive masses

from PS and subsequent symmetry breaking, given only the renormalizable Lagrangian

– 13 –

Page 15: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

terms above. For these fields, either the bilinear mass term or effective masses induced by

higher-dimensional operators play an important role and must be included as independent

parameters. From a phenomenological viewpoint, we are mainly interested in the lowest

possible mass for each of those fields. We discuss this issue in section 7.1.

While some multiplets acquire masses proportional to either one of the symmetry-

breaking scales vT , vΣ, vΦ, there are various cases where the mass becomes of order v2Φ/vT

or v2Φ/vΣ, which can be significantly smaller than vΦ if there is a hierarchy between the

vevs. In other words, there is an extra see-saw effect, unrelated to the well-known neutrino

see-saw [27, 42, 43]. We denote this induced mass scale by MIND. It is located below

the scale where PS is completely broken down to the MSSM symmetry group. A generic

expression is

MIND ∼v2

Φ

vΣ + vT. (5.1)

Depending on the model class, some of the field multiplets F , h, or TL become asso-

ciated with MIND7 (cf. table 2). We thus get “light” supermultiplets consisting of scalars

and fermions, which may be colored, charged, or neutral, and acquire a mass that does not

coincide with either of the symmetry-breaking scales. If the hierarchy between the vevs is

strong, MIND may be sufficiently low to become relevant for collider phenomenology. In

model classes B, C, and F, it provides a µ term for h and may thus be related to electroweak

symmetry breaking. In any case, the threshold MIND must be taken into account in the

renormalization-group running of the gauge couplings.

5.2 Goldstone bosons

Not all of the scalar fields are physical: since the broken symmetries are gauged, nine

of the scalar fields are Goldstone bosons that provide the longitudinal modes of the

PS gauge bosons that are integrated out in the breaking down to the MSSM. Six of

them come from SU(4)C → SU(3)C (5.2a), (5.2b), and two additional ones implement

SU(2)R → U(1)R (5.2c), (5.2d). The last one comes from the breaking of the U(1) sub-

groups U(1)B−L ⊗U(1)R → U(1)Y (5.2e). We identify these Goldstone bosons as

GB1,2,3 = −i

√3

2

vΣΦR

3 − i

√3

2

vΣΦ∗R

3 + Σ3 + Σ∗3 , (5.2a)

GB4,5,6 = i

√3

2

vΣΦ∗R

3+ i

√3

2

vΣΦ

R3 + Σ3 + Σ∗

3 , (5.2b)

GB7 = TR11

+ T ∗R11− i√

2

vTΦ

R11− i√

2

vTΦ∗R

11, (5.2c)

GB8 = TR1−1

+ T ∗R1−1

+i√2

vTΦR

1−1+

i√2

vTΦ∗R

1−1, (5.2d)

GB9 = Im(ΦR

10

)− Im

R10

). (5.2e)

Here 3 and 3 are the (3,1) 23

and (3,1) 23

components of the Higgs fields ΦR, ΦR and Σ,

respectively. For vanishing vevs, the corresponding fields do not mix into the GBs. Thus,

if vΣ = 0 (vT = 0), the Goldstone bosons GB1−6 (GB7/8) are only mixtures of Φ.

7In class F, this applies also to the singlet part of TR.

– 14 –

Page 16: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

5.3 MSSM Higgs

Apart from the matter fields and Goldstone bosons, the chiral superfield spectrum must

provide the Higgs bi-doublet of the MSSM that is responsible for electroweak symmetry

breaking. Above the TeV scale, this multiplet should appear as effectively massless; mass

terms (which together set the EWSB scale) are provided by soft-SUSY breaking parameters

and the µ term which mixes both doublets. We note that the µ term may result from the

vev of one of the electroweak singlets in the model, i.e., the model may implement a

NMSSM-type solution for the µ problem [44, 45].

The obvious candidate for the MSSM Higgs bi-doublet is the superfield h. From table 2

we read off that the PS-breaking contribution to the h mass is see-saw suppressed in models

with vT 6= 0 (B, C, D and F). In these models, the electroweak hierarchy is generated, at

least partly, by a high-energy hierarchy in the PS symmetry-breaking chain.

However, the ΦL multiplets provide further Higgs candidates. In particular, while the

right-handed doublets in ΦR serve as Goldstone bosons for the SU(2)R breaking and are

thus unphysical, the mirror images in ΦL are to be considered as physical scalars (above

the EWSB scale). Above the L-R breaking scale, these fields are protected against mass

terms, since all contributions to the mass are absorbed in the minimization conditions.8

Therefore they can only acquire nonzero mass due to mixing effects proportional to lhΦ

and lTΦ breaking LR-symmetry. With respect to the SM gauge symmetry, they have the

same quantum numbers as h, and will be called hΦ in the following.

In the limit vΦ � vT , the mixing effect which provides a mass for hΦ becomes negligible,

while there may still be an large contribution to the h mass. For the h/hΦ system, we obtain

a mass matrix with an approximate eigenvalue structure

µ ≈ mh +l2hΦ

lTΦ

v2Φ

vT, (5.3a)

m′hΦ≈ lTΦ vT , (5.3b)

where we allow for mh as an independent µ term, not directly related to PS breaking. Both

mass terms may be as low as the TeV scale where soft-breaking terms come into play. In

other words, the MSSM Higgs bi-doublet (in particular, the Higgs boson that has been

observed) may belong to either h or hΦ, or be a mixture of both.

If vT vanishes (class E), the situation becomes more complicated. Now, both masses

get a contribution of the order lhΦ vΦ. For mh = 0, both mass eigenvalues are degenerate.

Conversely, if the mixing between h and hΦ vanishes, they are maximally split (mh, 0). For

small mixing (lhΦmh � vΦ), we get an additional see-saw effect and a factor l2hΦv2Φ/mh for

the smaller eigenvalue, which generates the effective µ term.

In short, in various classes of PS models, the MSSM Higgs bi-doublet may be naturally

light, and it could actually originate from the ΦL superfields.

8The mass of ΦR is fixed since it is proportional to the vev which we keep fixed.

– 15 –

Page 17: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

5.4 The F multiplet

The mass of the multiplet F is generically see-saw suppressed and thus comparatively

light. This appears as a common feature of all model classes with more than one scale. It

can couple to matter via WYukawa. Since F is a SU(4)C antisymmetric tensor, the possi-

ble Yukawa couplings provide both diquark and leptoquark couplings, explicitly breaking

baryon number in the low-energy theory. In fact, F is the analog of the colored Higgs field

which in SU(5) GUT models induces rapid proton decay unless it is very heavy.

However, in PS models the Yukawa matrices YF and Y are not related, so there is

no doublet-triplet splitting problem [21, 22]. Hence, by omitting the coupling of F to Ψ,

proton decay is excluded in the presence of all gauge symmetries. This can be achieved,

for instance, by a flavor symmetry or by an appropriate discrete quantum number.

If the F multiplet is sufficiently light, it may provide detectable new particles at col-

liders. Without the YF Yukawa coupling, there is no immediate decay to MSSM matter

fields, but other terms in the Lagrangian provide indirect decay channels. In this situation,

the particles (color-triplet scalars and fermion superpartners) may become rather narrow

as resonances.

5.5 SM singlets

The most complicated mass matrix belongs to the electroweak singlets that are contained

in the various PS multiplets. Even if we do not consider PS singlets, there are still five

electroweak singlets which can mix non-trivially. In the general case, their masses cannot

be calculated in closed analytical form. To get a handle on these particles, we computed the

dependence on the different scales numerically. Given one of the scale hierarchy patterns

introduced above, we find additional scales and new hierarchy patterns which may have

interesting consequences for flavor and Higgs physics. However, singlets do not contribute

to the running of the gauge couplings at leading-logarithmic level, so we do not attempt a

detailed discussion of the singlet sector in the present work.

5.6 Matter couplings

The renormalizable superpotential contains terms that couple h to matter Ψ, so matter

fields can get masses via the h vev, after electroweak symmetry breaking. However, there

is no reason for flavor physics to originate solely from the renormalizable superpotential.

In particular, if hΦ turns out to be the MSSM Higgs, there are no contributions from this

superpotential at all.

Instead, we expect significant and non-symmetric contributions to masses and mix-

ing from non-renormalizable terms that induce effective Yukawa couplings at one of the

symmetry-breaking scales. The generated terms are proportional to powers of the various

vevs in the model, and suppressed by masses of heavier particles, by MGUT, or MPlanck.

There is ample space for flavor hierarchy in the resulting coefficients. For instance, if we

consider dimension-four terms in the superpotential, we identify the following interactions

– 16 –

Page 18: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

that can affect matter-Higgs Yukawa couplings in the low-energy effective theory,

WNLOYukawa =

YhΣ

ΛΨL hΣΨR +

ΛΨL

(ΦLΦR + ΦLΦR

)ΨR

+YEF

ΛΨLEF ΨR . (5.4)

There are additional couplings to gauge singlets, which may also be flavored, so overall

there is great freedom in assigning masses and generating hierarchies in the mass and

mixing parameters.

5.7 Neutrino mass

As a left-right-symmetric extension of the SM, PS models contain right-handed neutri-

nos and allow for a Dirac neutrino mass term. Furthermore, the extra fields that are

present in our setup can induce any of the three neutrino see-saw mechanisms for mass

generation [46, 47]. The fields ΦR and TR provide singlets with a vev that can couple

to right-handed neutrinos, generating a Majorana mass term proportional to vΦ and vT ,

respectively. Combined with the Dirac mass, this results in a type I see-saw. The field

TL contains SU(2)L-triplet scalars and fermions and may thus induce a type-II or type-III

see-saw mechanism.

6 Unification conditions

Within the framework of PS model classes that we have defined in the previous sections,

we now impose unification conditions on the gauge couplings. As stated above, we require

complete unification for all gauge groups to a GUT symmetry that contains SO(10). The

unification scale MGUT where this should happen is not fixed but depends on the spectrum.

At each threshold below this where the spectrum changes, i.e., particles are integrated out,

we state the appropriate matching conditions.

We work only to leading-logarithmic level, where non-abelian running gauge couplings

are continuous in energy, and matching conditions depend only on the spectra.9 Abelian

gauge couplings do exhibit discontinuous behavior as an artifact of differing normalization

conventions in different effective theories.

The leading-logarithmic running of a gauge couplings between fixed scales µ1 to µ2 is

given by

1

αi(µ2)=

1

αi(µ1)− bi

2πln

(µ2

µ1

). (6.1)

Here, the bi are group theoretical factors that can be calculated from the representations

of the particles [48].

9At next-to-leading order, the superpotential parameters enter the running. However, given the great

freedom in choosing a model in the first place, there is little to be gained from including such effects in our

framework.

– 17 –

Page 19: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Inserting the intermediate mass scales (case B for definiteness) as discussed above, the

complete running is a sum of multiple terms,

1

αi(MGUT))=

1

αi(MZ)−b(1)i

2πln

(MSUSY

MZ

)−b(2)i

2πln

(MIND

MSUSY

)−b(3)i

2πln

(vΦ

MIND

)−b(4)i

2πln

(vTvΦ

)−b(5)i

2πln

(vΣ

vT

)−b(6)i

2πln

(MGUT

), (6.2)

where MIND denotes the additional see-saw induces scale introduced in (5.1). Since this

scale depends on the numerical values of superpotential parameters, which we do not

determine, we treat this as a free parameter. Distinguishing different model classes with

their corresponding hierarchy patterns, we have to appropriately adapt the ordering of

scales and the definition of the b(n)i .

The calculation of the coefficients in this formula is straightforward and can be found in

appendix A. For simplicity, we always assume that the listed scales exhaust the available

hierarchies, and no further hierarchies from couplings become relevant here. Thus, we

assume all additional scalar fields to be integrated out at their “natural” mass scale, which

is determined by the considerations in the previous section.

In passing, we note that the calculation is actually independent from a supersymmetry

assumption. The supersymmetric and non-supersymmetric frameworks differ only in the

form of the β-function (cf. (A.5), (A.6)).

Regarding U(1) couplings with their normalization ambiguity, we have to explicitly

consider the unification condition for

U(1)R ⊗U(1)B−L −→ U(1)Y . (6.3)

To define the hypercharge coupling strength, we explicitly calculate the unbroken di-

rection and identify the charges of the larger groups. This results in a relation between the

group generators and therefore between charges and couplings. We obtain10

Y =B − L

2+ TR

3 , (6.4a)

α−1Y (vΦ) =

2

3α−1B−L(vΦ) + α−1

R (vΦ) . (6.4b)

For the non-abelian symmetry breaking steps, the unification conditions just depend

on the breaking pattern, i.e.,

GUT −→ SU(4)C ⊗ SU(2)L ⊗ SU(2)R , (6.5a)

SU(4)C −→ SU(3)C ⊗U(1)B−L , (6.5b)

SU(2)R −→ U(1)R , (6.5c)

10We do not rescale U(1) in order to match the SU(5) normalization, as is often done in the literature.

– 18 –

Page 20: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

where the GUT group contains SO(10). These three breaking patterns result in the match-

ing conditions11

α−14 (MGUT) = α−1

L (MGUT) = α−1R (MGUT) ≡ α−1

GUT(MGUT) , (6.6a)

α−13 (vΣ) = α−1

B−L(vΣ) ≡ α−14 (vΣ) , (6.6b)

α−1U(1)R

(vT ) = α−1R (vT ) = α−1

L (vT ) (6.6c)

respectively.

In addition to the unification and matching conditions, we have the additional con-

straint that the mass scales are properly ordered. For instance, for class B we have:

MSUSY ≤MIND ≤ vΦ ≤ vT ≤ vΣ ≤MGUT . 1019 GeV. (6.7)

Furthermore, the coupling strengths αi have to be sufficiently small and positive at all

mass scales, so we do not leave the perturbative regime.

Counting the number of conditions and free parameters (scales), we observe that the

models are still under-constrained. Hence, we can derive constraints for the mass scales

and exclude particular models, but not fix all scales completely. Nevertheless, imposing

unification does restrict the model parameter space significantly, as we can show in the

following sections.

7 Supersymmetric Pati-Salam models

In this section, we study the supersymmetric models that are consistent with our set

of assumptions. We scan over all models by varying the number of superfield generations

(0,1,3) that are present in each effective theory (i.e., between the various symmetry breaking

scales), independently for each gauge multiplet.

We force the masses of all superfields that are not part of the low-energy (MSSM) spec-

trum, to coincide with one of the thresholds MIND, vT , vΦ, etc., as explained in section 5.

In any case, order-one prefactors in the mass terms would only enter logarithmically in the

gauge-coupling unification (6.2) and matching conditions (GCU) (6.4b), (6.6), (6.7). This

is a minor uncertainty. We should note, however, that additional coupling hierarchies, as

they exist in the flavor sector of the known matter particles, are also possible and lead to

a wider range of possibilities which we do not investigate further.

With these conventions understood, the model scan will be exhaustive, since we vary

just discrete labels. In total, there are 1078 distinct configurations. Since we now are

dealing with mass scales rather then vev structures or specific mass eigenvalues, we change

our notation from vevs to mass scales (see appendix B).

For each model, we analytically calculate the numerical values of mass scales for which

the GCU conditions stated in the previous section can be fulfilled. Models where no solution

is possible are not considered further. For the remaining models, we obtain model-specific

relations between the mass scales. As a result, we can express those scales as functions

11In case one of the breaking steps is absent, the corresponding conditions apply at the next step below.

– 19 –

Page 21: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

of one or two independent mass parameters that we may choose as input. Varying those

within the possible range, we obtain allowed ranges for all mass scales, within each model

separately.

For all numerical results, we fix the common soft SUSY-breaking scale at MSUSY =

2.5 TeV.12

7.1 Extra mass parameters

As discussed in the previous section, in some model classes the fields E, Σ and T do

not necessarily obtain a mass term from symmetry breaking. Thus, their masses must

be treated as extra free parameters. To get a handle on these scales, we considered all

possibilities for assigning the mass scales of these superfields to the other mass scales

in our framework, while keeping the GCU conditions. As a result, we can exclude the

possibility that these extra scales are at the lower end of the spectrum.

More specifically, in all model classes we find a lower bound for the colored E multiplet

of about mE & 108 GeV. Similar results apply to Σ and T , if we do not consider lowering

the GUT scale below about 1016 GeV. In other words, the proton stability constraint which

limits the GUT scale, suggests that these fields have rather large masses. For the further

scan over models, we fix their mass scales, whenever they are not determined by the vevs,

at MPS.

Any model has to provide a candidate for the electroweak Higgs boson. This excludes

a large invariant mass for the corresponding PS multiplet. We therefore do not include an

explicit mass term for the h superfield. We have checked to what extend hΦ may serve as

the low energy MSSM-like Higgs. This is possible, but only for class E. Thus, we must

include at least one generation of h in the classes B, C and F. This reduces our scan to

828 configurations.

7.2 General overview

Before we discuss the various classes and types of models in more detail, we summarize

generic features and specific observations that we can extract from the study of all 828

supersymmetric models.

Roughly one half of all models are capable of GCU. Except for class E, all such

configurations exhibit a unification scale MGUT > 1016 GeV and are thus favored by the

non-observation of proton decay. In class E this is true for half of them.

In contrast to the classes C, E and F, the allowed ranges in class B are rather con-

strained, so in this class, models can be fixed in a semi-quantitative way.

We now take a look at the low energy spectrum of the different classes. These can be

extra color triplets (F ) or SU(2)L triplets (TL). In particular, we are interested in models

where the lowest new threshold MIND is already in the TeV-range, so new particles are

potentially accessible at colliders, while full unification occurs near the Planck scale. 114

models satisfy these conditions. 72 of them are categorized as class E and as such contain

12We also considered a lower SUSY-breaking scale of MSUSY = 250 GeV which is disfavored by LHC

data; it turns out that the unification conditions are generically easier to satisfy for the larger value of the

soft SUSY-breaking scale.

– 20 –

Page 22: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Figure 2. Graphical illustration of the lowest new mass scale, dependent on the multiplicity in

the low-energy spectrum. We vary the number of low-lying SU(2) triplets T (x axis) and low-lying

color triplets F (y axis), independently. The colors indicate the lowest mass scale, ranging from

green (SUSY scale) to red (Planck-scale). White6 squares correspond to configurations not leading

to GCU or which are inconsistent with the class definitions.

only light color triplets. 34 of them are categorized as class C. In class B, only a few

models fulfill this condition, none in class F.

One key feature of our configurations is that we allow for up to six light SU(2)Lscalar bidoublets h and hΦ (three each). Most configurations with more than one fall in

class E. But also in class C there is a handful of configurations. In class E most successful

configurations have more than one light bidoublet. This is because h and hΦ are taken

as degenerate in mass. A more detailed discussion will follow when we look explicitly at

class E.

In class E the LR-breaking scale is allowed to be rather light. We find 120 distinct

models with MLR < 105 GeV. Also in class C we find some configurations.

As a generic observation, SU(2) triplets T rarely get low mass, and, if present in the

intermediate range, tend to be associated with lower GUT scales. One exception is class F

with three generations of light triplets F . Here, light T are realized for a GUT scale

near MPlanck. Light color triplets F , i.e., extra quarks and their superpartners, are more

common. Actually, in class E they are allowed over a large mass range for the GUT scale.

In classes A to D, we may have color triplets around some 100 TeV, as long as there is only

one generation of light SU(2) triplets. In class F, the fields T and F are generically more

heavy (cf. section 7.4).

We illustrate these results in figure 2. The figure displays a considerable fraction of

models where new matter is possible at the lowest scales (green squares), so we should

be prepared to observe exotic particles, or at least their trace in precision observables, at

collider experiments.

Another observable of interest is the preferred mass range for right-handed neutrinos.

In our setup, the Majorana mass parameter should be of the order 〈Φ〉 where all symmetries

– 21 –

Page 23: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

class B class C class E class F∑

scanned 144 144 324 216 828

GCU 18 57 254 29 358

MGUT > 1016 GeV 18 57 131 29 235

MIND < 10 TeV and MGUT > 1016 GeV 8 34 72 0 114

MLR < 100 TeV 1 11 108 0 120

1012 GeV < MNR< 1014 GeV 16 42 123 3 184

MIND ∈ [0.1, 10]v2Φ

vΣ+vT14 20 203 26 263

Table 3. Number of configurations full filling certain conditions.

that protect this term are broken. Scanning all configurations with respect to this scale,

we find 184 with 1012 GeV . MNR. 1014 GeV. Actually, in classes B, C, three-quarter

and in class E still half of all successful models fall in this category. Only in class F, this

scale is typically higher; only 10% of the models result in a value in this range.

We also find that a neutrino mass scale in this range is associated with at most one

generation of h fields. As an interesting non-standard scenario, hinting at E6 grand unifica-

tion, three light h generations are also possible, but accompanied by heavier right-handed

neutrinos. In classes B and C, we find a few successful models with three h generations,

and none in class F. However, there is quite some space for this scenario in class E.

So far, we treated MIND as a free parameter. To obtain more specific predictions, we

imposed the restriction MIND ∈ [0.1, 10]v2Φ

vΣ+vT. While this does not significantly reduce

the number of allowed models, it drastically reduces the configurations with TeV-scale new

particles. Most of these models still allowing TeV-scale new particles belong to class E. In

this class, one quarter of all models allows for colored triplets below about 10 TeV.

In some models, the GCU constraints pin down all scales to a narrow range. The most

obvious case is standard SO(10) coupling unification at the GUT scale, i.e., all vevs are

of the same order of magnitude and located at MGUT. Clearly, this well-studied model is

contained in our scan as a limiting case. We reproduce the observation that for this case,

the only light multiplet is the MSSM Higgs h. However, we also find a few models where

scales are essentially fixed, but the spectrum and unification pattern is different. Those

are all classified as class E. More generic statistics can be read off table 3.

7.3 Class E

From the overview above we can conclude that class E contains the largest set of models

with phenomenologically interesting features. In this class, the ordering of new thresholds

is, in ascending order: the scale of soft SUSY breaking MSUSY, the see-saw induced scale

MIND, the left-right unification scale MLR, the scale where Pati-Salam symmetry emerges

MPS, and the scale of complete gauge-coupling unification MGUT.

E: MSUSY ≤MIND ≤MLR ≤MPS ≤MGUT (7.1)

– 22 –

Page 24: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Two of those scales can be regarded as free parameters; we may take the lowest (MIND)

and highest (MGUT) scale for that purpose. The other scales are then fixed by the matching

and unification conditions, if they can be satisfied at all.

There are 324 class-E models. In 254 configurations it is possible to implement gauge-

coupling unification (GCU), i.e., satisfy all matching and unification conditions. 131 of

these configurations allow for a scale MGUT > 1016 GeV. 76 configurations are able to

produce GCU at the Planck scale.

As discussed in the previous section, in class E the superfield h is not necessarily

contained in the spectrum. We found 77 configurations leading to GCU in which the

MSSM-like Higgs is hΦ. As mentioned above, most of the configurations have more than

one bidoublet at the EWSB scale. Only 29 feature exactly one. There are roughly 50 sets

with 1, 2 or 6 bidoublets each and 81 with 4. Zero or five are generally excluded by our

setup. We also find that a larger number of light bidoublets is correlated with lower scales.

Especially in the case of six bidoublets, we find that the maximal value for the GUT scale

is MGUT < 1016 GeV.

Note that in some models of class E, the see-saw scale MIND is not phenomenologically

relevant (cf. table 2), since they do not contain the colored superfield F . Thus, we should

break down the model space according to the multiplicity of the F multiplet: zero on the

one hand (no see-saw scale), one or three on the other hand.

If there is no field F , the lowest-lying threshold above the soft SUSY-breaking scale

is the scale of left-right-handed unification, MLR. It turns out that, in some models, this

scale can be as low as the SUSY scale. At the other end of the spectrum, the complete

unification scale MGUT can vary in the range between 109 GeV and 1019 GeV.

In the cases of one or three generations of F , the see-saw scale can be as low as the

soft SUSY-breaking scale, independent of the GUT scale. The upper bound for the see-saw

scale is only fixed by the requirement that it is the lowest-lying scale, and is approximately

MIND . 1016 GeV.

As mentioned before, there are configurations fixing all scales. These posses three

generations of h, Φ, TL/R and one generation of Σ. The multiplicity of E is not fixed. For

three generations of E also one or zero generations of TL/R are possible. The LR-scale is in

these configurations fixed to MLR = 7× 103 GeV and the PS scale to be MPS = 109 GeV.

Let us now consider in somewhat more detail, the three particular model types de-

scribed in section 3.

Type Em: minimal model. In class E, the minimal model is the standard MSSM

without Higgs,13 supplemented only by the additional fields Φ and Σ above their respective

thresholds. Looking at table 2, we see that this setup does not provide a see-saw scale.

Hence, the sub-unification scales depend only on one free parameter, which we take to be

MGUT. Figure 3(a) shows the variation of the three other scales as a function of MGUT.

For the value of MGUT ≈ 3× 1016 GeV, all scales approximately coincide. This is the

minimal MGUT value for which GCU is possible in this setup. For this particular parameter

point, the GUT symmetry (e.g., SO(10)) directly breaks down to the MSSM by virtue of

13The electroweak Higgs is contained in ΦL.

– 23 –

Page 25: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

13 14 15 16 17 18 19

Log@MGUTD@GeVD

4

7

10

13

16

19

Log@MD@GeVD

(a) Possible scale variation leading to GCU. The

GUT-scale is shown in black, the PS-scale in blue

and the LR-scale in red. A IND-scale is not present

in this type. The dots indicate the scales for the

exemplary plot shown in (b).

100 105 108 1011 1014 1017

MGUT

@GeVD0

20

40

60

80

100Α-1

(b) Exemplary running of the gauge couplings for

complete unification at MGUT = 1018 GeV. The

hypercharge coupling is shown in red, the B − L in

green, the weak in blue and the strong coupling in

black.

Figure 3. Variation of the unification scales and exemplary running of the gauge couplings for the

type Em.

vΣ = vΦ, so this is actually the standard SO(10) scenario. If we demand a larger GUT

scale, the Pati-Salam scale decreases but never drops below MPS & 1014 GeV. The left-

right unification scale can vary in the range 1011 GeV . MLR . 1016 GeV. This is within

the favored mass range for right-handed neutrinos. A sample unification plot is shown in

figure 3(b).

Type Es: SO(10)-like models. We now turn to a model with complete SO(10) rep-

resentations below the GUT scale. With this spectrum, we can vary independently MIND

and MGUT, within a certain range.

It turns out that MGUT, in this type of model, cannot reach the Planck scale. The

maximal allowed value for MGUT depends on MIND and decreases with increasing MIND.

The value of MIND, and thus the mass of the color-triplet fields F , can be as low as the

soft SUSY-breaking scale.

Another important difference is that the scales approach each other when MGUT gets

larger. In figure 4, we plot the variation of the sub-unification scales as function of MGUT

for three fixed values of MIND (solid MIND = 103.4 GeV, dashed MIND = 105.4 GeV and

dotted MIND = 107.4 GeV). For the lowest value of MIND, it is possible to have GCU

without any sub-unification. For larger MIND we see a gap opening between MLR and

MPS, but it is still possible to achieve MPS = MGUT.

Type Ee: E6-inspired models. In this model, we combine complete SO(10) multiplets

with three generations of the “MSSM-like” Higgs fields h and the color-triplets F , so within

each generation, matter fields unify with the MSSM Higgs fields and an additional singlet

each, to complete 27 representations of E6. In this scenario, GCU is possible over a wide

range of mass scales.

– 24 –

Page 26: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

13 14 15 16 17 18

Log@MGUTD@GeVD

4

6

8

10

12

14

16

18

Log@MD@GeVD

(a) Possible scale variation leading to GCU. The

GUT-scale is shown in black, the PS-scale in blue

and the LR-scale in red. The variation in the IND-

scale is shown discrete with MIND = 103.4 GeV

as solid lines, MIND = 105.4 GeV dashed MIND =

107.4 GeV dotted. The dots indicate the scales for

the exemplary plot shown in (b).

100 105 108 1011 1014 1017

MGUT

@GeVD0

20

40

60

80

100Α-1

(b) Exemplary running of the gauge couplings for

complete unification at MGUT = 1015.1 GeV. The

hypercharge coupling is shown in red, the B − L in

green, the weak in blue and the strong coupling in

black.

Figure 4. Variation of the unification scales and exemplary running of the gauge couplings for the

type Es.

Like in the model Es discussed above, the separation between the sub-unification scales

decreases with increasing scale MGUT. Over the whole range of MIND it is possible to have

the PS and GUT unification coincide, MPS = MGUT. Complete unification at a single

scale is possible for MGUT ≈ 1016.4 GeV if the scale of light triplets is equal to the soft

SUSY-breaking scale, MF = MSUSY = 2.5 × 103 GeV. This is the well known SU(5)

limiting case, since all fields of the low energy spectrum can be grouped to complete SU(5)

representations.

Compared to the previous two model types, the gauge coupling at the unification point

α−1GUT is significantly lower and, in some cases, touches the non-perturbative regime. In

figure 5, we show the variation of scales and an exemplary unification plot.

7.4 Class F

This model class has a more restricted phenomenology. Nevertheless, this class contain

some models that exhibit GCU.

In class-F models, SU(2)R (and thus LR symmetry) is broken at vT , above the scale vΦ

where SU(4)C reduces to color. We therefore might expect closer relations between lepton-

flavor and quark-flavor mixing. The relevant scales of this class are, in ascending order:

the see-saw induced scale MIND, the quark-lepton unification scale MQL, the Pati-Salam

scale MPS, and the unification scale MGUT.

F: MSUSY ≤MIND ≤MQL ≤MPS ≤MGUT (7.2)

Table 2 indicates that all models in this class do have the additional see-saw scale

MIND. The intermediate scales tend to be higher than in class E above.

– 25 –

Page 27: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

13 14 15 16 17 18

Log@MGUTD@GeVD

4

6

8

10

12

14

16

18

Log@MD@GeVD

(a) Possible scale variation leading to GCU. The

GUT-scale is shown in black, the PS-scale in blue

and the LR-scale in red. The variation in the IND-

scale is shown discrete with MIND = 103.4 GeV

as solid lines, MIND = 105.4 GeV dashed MIND =

107.4 GeV dotted. The dots indicate the scales for

the exemplary plot shown in (b).

100 105 108 1011 1014 1017

MGUT

@GeVD0

20

40

60

80

100Α-1

(b) Exemplary running of the gauge couplings for

complete unification at MGUT = 1015.1 GeV. The

hypercharge coupling is shown in red, the B − L in

green, the weak in blue and the strong coupling in

black.

Figure 5. Variation of the unification scales and exemplary running of the gauge couplings for the

type Ee.

Out of the 216 class-F models only 29 configurations are consistent with GCU. In all

cases, MGUT can be as large as the Planck scale.

As discussed in the overview, light degrees of freedom are not possible in this class.

As above, we break down the set of configurations according to their content of light

fields. There are now 6 categories. In addition to the ones mentioned above (zero, one, or

three generations of F ), we have to distinguish cases of one or three generations of SU(2)

triplets TL.

The minimal MIND value is strongly dependent on the number of SU(2) triplets. In

the case of three triplets, it is strictly larger than 1016 GeV, essentially independent of the

number of F fields. Thus, let us look at the configurations with only one TL/R generation.

In these configurations, MIND is bound to be larger than MIND & 106 GeV. It is

realized for three generations of F and rises the less are included.

We conclude that in class F, the extra fields may play a role for flavor physics in an

intermediate energy range, but are unlikely to be observable in collider experiments.

Type Fm: minimal model. The minimal model of class F contains the superfields Φ

and TL/R in addition to the MSSM spectrum.

In models of this type, the lowest possible see-saw mass value is MIND ≈ 1012 and

LR-scale is MLR & 1015. Thus these are ruled out, since the mass of the EWSB Higgs is

associated to the LR-scale. A next to minimal setup explicitly including one generation of

h is not able to produce GCU.

Type Fs/Fe. In type Fs, there is no model consistent with GCU. This is because α−13

grows to fast and overshoots α−12 before the condition (6.4b) for a possible QL-scale can

be fulfilled. A model of type Fe consistent with GCU is also not possible.

– 26 –

Page 28: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Type Ff: flavor-symmetry inspired model. In the absence of the previous types,

we take a look at a configuration which might be viewed as E6-inspired, but with the

additional condition that two of three Higgs bidoublets get heavy (Planck scale) by some

unspecified mechanism. In this model, unification is possible over a wide range of mass

scales. There is a strong correlation of MIND and MGUT, so the latter is the only relevant

parameter. MIND can vary between 107 GeV . MIND . 1016 GeV. For its largest allowed

value all scales are approximately equal, which is the SO(10) limiting case. Conversely, the

lowest possible MIND value corresponds to GUT unification near the Planck scale. The

QL, the PS and the GUT scales are nearly degenerate in any case.

As mentioned above, there is no possible configuration leading to GCU with three

generations of the field h.

7.5 Classes A to D

In these classes, we effectively combine model classes E and F. There are five different

scales, two of which are fixed by requiring GCU. For concreteness, we also fix MGUT =

1018.2 GeV, i.e., we assume complete unification at the Planck scale. Still, we can choose

two parameters independently, so we will obtain allowed and forbidden regions, but no

one-to-one correspondences.

More specifically, we distinguish the cases vT ≤ vΣ (class B) and vT > vΣ (class C),

where A and D appear as limits. The ordering of scales in the two scenarios is

B: MSUSY ≤MIND ≤MU1 ≤MLR ≤MPS ≤MGUT , (7.3)

C: MSUSY ≤MIND ≤MU1 ≤MQL ≤MPS ≤MGUT . (7.4)

Here, MU1 indicates the mass scale where the extra U(1) groups break down to hyper-

charge. This is also the natural scale for a mass term of the right-handed neutrinos. The

spectrum below MU1 still contains the extra particles that are integrated out at the lower

see-saw scale MIND, which in turn is located above the soft SUSY-breaking scales. The

labels LR and QL refer to left-right and quark-lepton symmetry breaking, respectively.

Of the 144 models in classes B and C, 18 (B) and 57 (C) are consistent with GCU,

respectively. We observe again that the number of TL generations has a strong impact

on phenomenology. First, we look at the case of three TL generations. In class C, MIND

depends on the number of generations of the field F . It ranges from MIND & 1015 GeV (no

F ) down to to MIND & 106 GeV (three F generations). The situation in class B is even

worse. Here GCU is not possible with less then three generations of the field F .

If there is only one generation of T , the MIND value can approach the SUSY scale,

independent on the number of F .

Type Bm/Cm: minimal model. In the minimal model, there is a single generation

of each of Φ, TL/R and Σ. We can achieve GCU for both Bm and Cm. The see-saw scale

is stuck at rather high values, MIND & 1013 GeV and MIND & 1010 GeV for type Bm and

Cm, respectively. Since we again do not explicitly add a EWSB Higgs, such models are

ruled out because of the nonexistence of light SU(2) doublets. If we include one generation

of h in addition, GCU is no longer possible.

– 27 –

Page 29: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

13 14 15 16 17 18

Log@MGUTD@GeVD

4

6

8

10

12

14

16

18

Log@MD@GeVD

(a) Possible scale variation leading to GCU. The

PS-scale is shown in black, the LR-scale in blue

and the MSSM-scale in red. The variation in the

IND-scale is shown discrete with MIND = 104 GeV

as solid lines, MIND = 107 GeV dashed MIND =

1010 GeV dotted. The dots indicate the scales for

the exemplary plot shown in (b).

100 105 108 1011 1014 1017

MGUT

@GeVD0

20

40

60

80

100Α-1

(b) Exemplary running of the gauge couplings for

complete unification at MGUT = 1018.2 GeV. The

hypercharge coupling is shown in red, the U(1)R in

brown, the B−L in green, the weak in blue and the

strong coupling in black.

Figure 6. Variation of the unification scales and exemplary running of the gauge couplings for the

type B199.

Types Bs/Cs and Be/Ce. These setups do not allow GCU.

Class-B/C models with MIND < 10 TeV. We may ask for models where the see-saw

scale is sufficiently low (say, MIND < 10 TeV) that the new particles can have an impact on

collider phenomenology. We find 8 (34) models where this is possible within class B (C),

respectively. One configuration with normal hierarchy is model B199,14 where we have

three copies of h, F and Σ and no E. In the inverted case there is a similar model C211,

which has the same spectrum, but three copies of Φ. The plots are shown in figure 6, 7.

8 Pati-Salam models without supersymmetry

We now turn to scenarios without supersymmetry. Obviously, there is much greater free-

dom for model building, if we ignore the naturalness problems that inevitably appears

when there are scalar fields in the spectrum. To limit this freedom in a meaningful way, we

consider the same classes of models as in the supersymmetric case, but omit the fermionic

superpartners of the additional multiplets. Analogously, we omit the scalar superpartners

of matter fields and the fermionic superpartners of gauge fields. This (ad-hoc) restriction

allows us to compare supersymmetric and non-supersymmetric models on the same basis.

The meaning of scales and symmetry breaking patterns are unchanged.

Since we found that in the non-SUSY case the resulting mass thresholds tend to be

lower, we fixed the GUT scale for the classes B and C to MGUT ≡ 1016 GeV. For the

value of MGUT = 1018.2 GeV which we used in the SUSY case, we would have only one

14For the meaning of numerical model indices, see appendix C.

– 28 –

Page 30: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

13 14 15 16 17 18

Log@MGUTD@GeVD

4

6

8

10

12

14

16

18

Log@MD@GeVD

(a) Possible scale variation leading to GCU. The

QL-scale is shown in black, the PS-scale in blue

and the MSSM-scale in red. The variation in the

IND-scale is shown discrete with MIND = 104 GeV

as solid lines, MIND = 107 GeV dashed MIND =

1010 GeV dotted. The dots indicate the scales for

the exemplary plot shown in (b).

100 105 108 1011 1014 1017

MGUT

@GeVD0

20

40

60

80

100Α-1

(b) Exemplary running of the gauge couplings for

complete unification at MGUT = 1018.2 GeV. The

hypercharge coupling is shown in red, the U(1)R in

brown, the B−L in green, the weak in blue and the

strong coupling in black.

Figure 7. Variation of the unification scales and exemplary running of the gauge couplings for the

type C211.

configuration (class C) for both classes that satisfy GCU. Lowering this scale even more,

the model space would be less constrained, but this is disfavored by the proton decay limits.

We considered the same set of 828 models, with the SUSY partners removed, as in the

previous section. Again, more than half of the models allow for GCU, given the modification

mentioned in the previous paragraph. Still it turns out that without SUSY, classes A–D

are disfavored but not excluded. In contrast to the supersymmetric case, a considerable

set of the successful models fall in class F. Again, the most belong to class E. In any case,

lower GUT scales tend to be favored.

As in the SUSY models discussed above, we observe a LR symmetry breaking scale

roughly around 1013 GeV in a large fraction of the successful models. Nevertheless, there

are models where this scale can be much lower, down to below 100 TeV in some cases.

In class E, there are again many models with the possibility for light color triplets F

in the TeV regime. In the non-SUSY setup, these are scalar particles and obviously cannot

mix with quarks. We have to assume that there are couplings of either leptoquark or

diquark type that render these particles unstable, originating from the PS-breaking sector.

Furthermore, in models of this kind there is a high probability that the MSSM Higgs is not

part of h but of the hΦ multiplet (see section 5.3). Similar to the SUSY case the number

of light bidoublets is constrained to be one in the classes A to D. For class E we found also

a similar result as in the SUSY case. Again, most configurations prefer four generations.

Six are somehow disfavored, and one to three are equally likely. In contrast to the previous

considerations, there are now plenty of configurations of class F with three generations

of bidoublets, but still one is in favor here. Again we see, that the multiplicity of these

bidoublets lowers the maximal unification scale.

– 29 –

Page 31: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

class B class C class E class F∑

scanned 144 144 324 216 828

GCU 10 30 230 201 471

MGUT > 1016 GeV 10 30 111 16 167

MIND < 10 TeV and MGUT > 1016 GeV 1 8 110 16 135

MLR < 100 TeV 0 0 136 0 136

1012 GeV < MNR< 1014 GeV 9 30 211 126 376

MIND ∈ [0.1, 10]v2Φ

vΣ+vT12 36 201 93 342

Table 4. Number of configurations full filling certain conditions in the non-SUSY case.

As in the SUSY case, we find that three generations of light SU(2) triplets T are

excluded. In particular, in class C the lower bound for those is MTL& 108 GeV. In class C,

the bound becomes 1011 GeV and in class B there is no GCU at all.

In the non-supersymetric case we again find in all classes a set of configurations fixing

all mass scales “exactly”. This usually corresponds to degenerate mass scales. A common

scale for such classes is MGUT ≈ 1014 GeV.

Again we find that fixing the induced scale MIND does not change the results very

much. Thus we see our assumptions justified to include this scale in our scans.

A general feature of non-SUSY spectra is the fact that the high-energy effective values

of the gauge couplings are larger than in the SUSY case. This is due to the lower number

of fields that contribute to the gauge-coupling running.

In the following, we do not repeat the detailed discussion of SUSY models but pick a

few selected models and model types with particular features. More generic statistics can

be read off table 4.

8.1 Class E

For class E it is possible to implement GCU in 230 configurations, of which 23 provide

a complete unification near the Planck scale. Similar to the supersymmetric case, we

find 88 configurations where the EWSB Higgs is provided by ΦL. On the other hand, an

interesting possibility is the existence of three Higgs generations (type Ee). Although, in

the non-SUSY case, there is no direct relation to E6 unification, we may take a look at such

models. In class E, we find that GCU is possible, and the GUT-scale can vary between

1014 GeV . MGUT . 1017 GeV. One possible configuration exhibits three generations of

F , one of Σ and Φ each, and no fields E or T . For this special configuration the possible

variation of the scales and the unification plot are shown in figure 8. Here we find that

the variation of the LR-scale strongly depends on the GUT-scale. The PS scale varies only

weakly and is always close to the GUT scale.

8.2 Class F

In class F, there are plenty of configurations leading to successful GCU. In general, we

find that there is not much scope for scale variation. The GUT scale can be as large as

– 30 –

Page 32: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

13 14 15 16 17 18

Log@MGUTD@GeVD

4

6

8

10

12

14

16

18

Log@MD@GeVD

(a) Possible scale variation leading to GCU. The

GUT-scale is shown in black, the PS-scale in blue

and the LR-scale in red. The variation in the

IND-scale is shown discrete with MIND = 104 GeV

as solid lines, MIND = 107 GeV dashed MIND =

1010 GeV dotted. The dots indicate the scales for

the exemplary plot shown in (b).

100 105 108 1011 1014 1017

MGUT

@GeVD0

20

40

60

80

100

Α-1

(b) Exemplary running of the gauge couplings for

complete unification at MGUT = 1015.8 GeV. The

hypercharge coupling is shown in red, the B − L in

green, the weak in blue and the strong coupling in

black.

Figure 8. Variation of the unification scales and exemplary running of the gauge couplings for the

non-SUSY type Ee.

MGUT ≈ 1017 GeV. The scales are close to each other, since the lightest scale is fixed to be

larger than MIND & 1013 GeV. In this class there are also models with GCU where all scales

are essentially fixed, and not far from the GUT scale. Those lead to MGUT ≈ 2×1014 GeV,

which is rather low. The LR-scale emerges between 1013 GeV ≤MLR ≤ 2× 1015 GeV.

One exemplary configuration leading to GCU above 1016 GeV is model F213. Here,

we have three generations of F and one of T . These scalar particles can be rather light,

potentially as low as the SUSY scale. In addition, this model contains three generations

of the fields Φ and Σ, and one generation of E. We show the possible scale variation and

a sample unification plot for this model in figure 9.

8.3 Class A to D

In class C, the QL-scale emerges typically close the PS scale. Likewise, the PS scale can

become as large as the GUT-scale, such that the energy range with pure PS symmetry may

vanish.

Looking at the possibility of three Higgs (h) generations, we do not find any configu-

rations in class B, and a few in class C. On the other hand, these model classes favor three

as the number of Φ generations.

In figure 10 and 11, we display the scale relations and gauge-coupling unification for

two distinct models B53 and C45.15 The former model contains three generations of F

which can be light. In addition it contains three generations of E and all other fields

ones. We found quite some range for scale variation. For the largest value of the PS scale

15For the meaning of numerical model indices, see appendix C.

– 31 –

Page 33: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

13 14 15 16 17 18

Log@MGUTD@GeVD

4

6

8

10

12

14

16

18

Log@MD@GeVD

(a) Possible scale variation leading to GCU. The

GUT-scale is shown in black, the PS-scale in blue

and the QL-scale in red. The variation in the IND-

scale is shown discrete with MIND = 104 GeV as

solid lines, MIND = 108.5 GeV dashed MIND =

1013 GeV dotted. The dots indicate the scales for

the exemplary plot shown in (b).

100 105 108 1011 1014 1017

MGUT

@GeVD0

20

40

60

80

100

Α-1

(b) Exemplary running of the gauge couplings for

complete unification at MGUT = 1016.5 GeV. The

hypercharge coupling is shown in red, the U(1)R in

brown, the weak in blue and the strong coupling in

black.

Figure 9. Variation of the unification scales and exemplary running of the gauge couplings for the

type non-SUSY F213.

13 14 15 16 17 18

Log@MGUTD@GeVD

4

6

8

10

12

14

16

18

Log@MD@GeVD

(a) Possible scale variation leading to GCU. The

PS-scale is shown in black, the LR-scale in blue

and the MSSM-scale in red. The variation in the

IND-scale is shown discrete with MIND = 104 GeV

as solid lines, MIND = 105 GeV dashed MIND =

106 GeV dotted. The dots indicate the scales for

the exemplary plot shown in (b).

100 105 108 1011 1014 1017

MGUT

@GeVD0

20

40

60

80

100

Α-1

(b) Exemplary running of the gauge couplings for

complete unification at MGUT = 1016 GeV. The

hypercharge coupling is shown in red, the U(1)R in

brown, the B−L in green, the weak in blue and the

strong coupling in black.

Figure 10. Variation of the unification scales and exemplary running of the gauge couplings for

the non-SUSY type B25.

(MPS ≈ 1015) we find a degeneracy of all lower scales. The induced scale can be as light

as some TeV.

Model C45 also features new particles at the TeV scale. Here we include three gen-

erations of Φ and Σ and all other fields ones. We again see quite some space to vary the

scales.

– 32 –

Page 34: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

13 14 15 16 17 18

Log@MGUTD@GeVD

4

6

8

10

12

14

16

18

Log@MD@GeVD

(a) Possible scale variation leading to GCU. The

QL-scale is shown in black, the PS-scale in blue

and the MSSM-scale in red. The variation in the

IND-scale is shown discrete with MIND = 104 GeV

as solid lines, MIND = 107 GeV dashed MIND =

1010 GeV dotted. The dots indicate the scales for

the exemplary plot shown in (b).

100 105 108 1011 1014 1017

MGUT

@GeVD0

20

40

60

80

100

Α-1

(b) Exemplary running of the gauge couplings for

complete unification at MGUT = 1016 GeV. The

hypercharge coupling is shown in red, the U(1)R in

brown, the B−L in green, the weak in blue and the

strong coupling in black.

Figure 11. Variation of the unification scales and exemplary running of the gauge couplings for

the non-SUSY type C27.

9 Summary of models

We have presented a survey of models with gauge-coupling unification along a path that

contains several intermediate scales, corresponding to left-right symmetry, quark-lepton

unification, Pati-Salam symmetry, and SO(10) or larger GUT symmetry. We studied both

supersymmetric and non-supersymmetric models, where the latter are derived from the

former by omitting all superpartners.

We have restricted the allowed new chiral superfields (or scalar fields, in the non-

SUSY case) below the GUT scale to a small well-defined set of multiplets, all of which

fit in small representations of SO(10) or E6. A large subset of the resulting models is

consistent with gauge-coupling unification, proceeding in several steps. In supersymmetric

models, there is slightly more freedom in varying the scales than in non-supersymmetric

models.

The assumptions and calculations do not constrain the model space in such a way that

we can get precise numerical predictions, but we can deduce characteristic patterns in the

scale hierarchies that correlate with specific choices for the spectrum.

In a wide range of models, GUT unification can be pushed up to the Planck scale.

This fact, together with the properties of Pati-Salam symmetry, significantly reduces the

strain that the non-observation of proton decay can put on GUT model building.

Additional thresholds in the intermediate range between observable energies and the

GUT scale are likely. Being associated with left-right or quark-lepton symmetry breaking,

they decouple flavor physics issues from the requirements of complete unification. We

have not considered flavor physics in any more detail, but expect that it can generically

be accommodated if non-renormalizable terms are properly included. Depending on the

– 33 –

Page 35: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

particular model and on the chosen set of flavor-dependent interactions, we expect specific

hierarchies, relations and predictions for the flavor sector.

As an extra feature, the (super)fields Σ and T (adjoint of SU(4)C and SU(2)R) and Φ

(fundamental of SU(2)L× SU(2)R) can cooperate to generate small mass terms for certain

particles, including SM-like Higgs doublets and new exotic (s)quarks, such that they can

be accessible at colliders. On the other hand, SU(2)L/R triplets as possible Pati-Salam

breaking Higgs fields tend to raise the LR symmetry scale above the scale of quark-lepton

unification and thus may enforce a direct relation between quark and lepton flavor physics.

Another generic property of the models under consideration is the scale of left-right

symmetry breaking, naturally associated with neutrino mass generation, in an intermediate

mass range. A neutrino mass scale significantly below the GUT scale is favored by numerical

estimates of see-saw mechanisms that can explain the small observable neutrino masses.

We also encounter models where gauge-coupling unification at high energies implies a

multi-Higgs doublet model at low scale, possibly with flavor quantum numbers. Further-

more, the observable Higgs doublet need not be a member of a (1, 2, 2) representation, as

often assumed, but can also originate from a (4, 2, 1) + c.c. representation, i.e., behave as

a scalar lepton. The effective µ term which sets the scale for low-energy Higgs physics,

can have a see-saw like form and thus be naturally suppressed with respect to the higher

symmetry-breaking scales.

10 Conclusions

In summary, we have studied a range of comparatively simple models that fit into the

framework of GUT theories with intermediate thresholds. We have taken a phenomeno-

logical viewpoint and specified the models in form of a chain of effective theories, as far as

we expect that a description in terms of weakly interacting four-dimensional gauge theory

can make sense.

It is remarkable that the most interesting phenomenology, which we may identify as

Planck-scale GUT unification, intermediate PS and LR scales, and new particles at TeV

energies, can be simultaneously realized in a number of distinct models (cf. appendix D).

This is not a generic feature. However, if not all of these conditions are to be satisfied

simultaneously, or allowing further multiplets or hierarchy patterns in couplings, the set of

interesting models becomes sizable.

One may consider fully specified GUT models that predict the appearance (or absence)

of a PS symmetry and the associated spectrum. However, the current lack of reliable

knowledge about strong and gravitational interactions which are expected at the highest

energies, denies attempts to ultimately favor or exclude alternative spectra and symmetry-

breaking chains.

In such a situation, it appears worthwhile to rather concentrate on the implications of a

sequence of new thresholds at intermediate energies, presumably in the context of PS sym-

metry. Our survey demonstrates that intermediate symmetry-breaking scales associated

with flavor mixing and mass generation can emerge in various energy regions, even if rather

specific and simple assumptions about gauge multiplets are imposed. The findings suggest

– 34 –

Page 36: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

that one should study the hierarchies within flavor observables in relation to hierarchies in

gauge-symmetry breaking, discuss both renormalizable and non-renormalizable operators,

and to combine gauge and flavor symmetries in a common framework which need not be

tied to ultimate GUT unification. This program, which has so far been pursued only for a

subset of the possible scenarios, deserves further efforts.

In summary, Pati-Salam models can easily accommodate unification in a multi-scale

framework. They provide a rich phenomenology and a promising background for new

approaches to the lepton and quark flavor problem.

A Beta-function coefficients

As stated in the paper (cf. section 6) the running of the gauge couplings can be described by

1

αi(µ2)=

1

αi(µ1)− bi

2πln

(µ2

µ1

). (A.1)

The coefficient bi of the running coupling can be calculated by means of the represen-

tation of the fields contributing at the given mass scale alone [48]. For each set of gauge

groups SU(N) with N ≥ 2 the contribution of an field with representation (I1, . . . , In) to

the running coefficient bi is given as

bRi = T (Ii)∏k 6=i

d(Ik) , (A.2)

where d(Ii) is the dimension and the normalization of the representation T (Ii). These can

be calculated using the representing matrices Ra

trRaRb = T (R) δab, (A.3)

with T (N) = 12 .

For a U(1) the contribution is up to an consistent rescaling:

bRU(1) = Y 2∏k

d(Ik) . (A.4)

The complete running coefficient depends now on whether we work in a supersymmetric

theory or not. For the non-supersymetric case one has to divide the fields in scalar and

fermionic contributions:

bSMi =

2

3

∑Rferm.

bRi +1

3

∑Rscalar

bRi −11

3C2(Gi) , (A.5)

where C2(Gi) = dim(Gi) is the quadratic Casimir operator.

Since in the supersymmetric case there is a superpartner for each scalar/fermionic field,

there is no need to divide the fields in such a way. Thus, the relation simplifies to

bSUSYi =

∑R

bRi − 3C2(Gi) . (A.6)

Table 5 displays the contribution of each field and its complete decomposition w.r.t.

the subgroups of PS symmetry.

– 35 –

Page 37: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Field PS LR SM bY bB−L b2 b3 b4

h (1, 2, 2) (1, 2, 2)0 (1, 2) 12

12 0 1

2 0 0

(1, 2)− 12

12

12 0

F (6, 1, 1) (3, 1, 1) 23

(3, 1) 13

13

43 0 1

2 1

(3, 1, 1)− 23

(3, 1)− 13

13

43 0 1

2

ΦR (4, 1, 2) (3, 1, 2) 13

(3, 1) 23

43

23 0 1

2 1

(3, 1)− 13

13 0 1

2

(1, 1, 2)−1 (1, 1)0 0 2 0 0

(1, 1)−1 1 0 0

ΦR (4, 1, 2) (3, 1, 2)− 13

(3, 1) 13

13

23 0 1

2 1

(3, 1)− 23

43 0 1

2

(1, 1, 2)1 (1, 1)1 1 2 0 0

(1, 1)0 0 0 0

ΦL (4, 2, 1) (3, 2, 1)− 13

(3, 2)− 16

16

23

32 1 1

(1, 2, 1)1 (1, 2) 12

12 2 1

2 0

ΦL (4, 2, 1) (3, 2, 1) 13

(3, 2) 16

16

23

32 1 1

(1, 2, 1)−1 (1, 2)− 12

12 2 1

2 0

Σ (15, 1, 1) (8, 1, 1)0 (8, 1)0 0 0 0 3 4

(3, 1, 1)− 43

(3, 1)− 23

43

163 0 1

2

(3, 1, 1) 43

(3, 1) 23

43

163 0 1

2

(1, 1, 1)0 (1, 1)0 0 0 0 0

E (6, 2, 2) (3, 2, 2) 23

(3, 2) 56

256

163

32 1 4

(3, 2) 16

16

32 1

(3, 2, 2)− 23

(3, 2)− 16

16

163

32 1

(3, 2)− 56

256

32 1

TR (1, 1, 3) (1, 1, 3)0 (1, 1)1 0 0 0 0 0

(1, 1)0 0 0 0 0 0

(1, 1)−1 0 0 0 0 0

TL (1, 3, 1) (1, 3, 1)0 (1, 3)0 0 0 2 0 0

Table 5. Full field content and breaking as well as all contributions to the beta function.

B Vacuum expectation values and mass scales

In the first part of this paper we calculate the superpotential and the masses of all su-

perfields. Therefore we use the vevs as natural scales. Since in this part the breaking

associated to the vev is not the most important thing but the fields they are related to, we

name them after those.

In the second part, we are primarily interested in the scales present in the running of

the gauge couplings. Hence, we switch our notation to the mass scales. These are labeled

– 36 –

Page 38: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

vev class B class C class E class F

vΣ MPS MQL MPS −vT MLR MPS − MPS

vΦ MMSSM MMSSM MLR MLRv2Φ

vΣ+vTMIND MIND MIND MIND

Table 6. Relation between the vevs and the mass scales for the different classes. In class A there

is no hierarchies in the vevs and thus all scales are equal to MPS. In class D there are only the

scales MPS and MLR.

Name #h #F #Φ #Σ #E #TL/R

SUSY models: Em 0 0 1 1 0 0

Fm 0 0 1 0 0 1

Es/Fs 1 1 1 1 1 1

Ee/Fe 3 3 3 3 1 1

Ff 1 3 1 1 1 1

Bm/Cm 0 0 1 1 0 1

Bs/Cs 1 1 1 1 1 1

Be/Ce 3 3 1 1 1 1

B199 3 3 1 3 0 1

C211 3 3 3 3 0 1

Non-SUSY models: E289 3 3 1 1 0 0

F213 1 3 3 3 1 1

B53 1 3 1 1 3 1

C45 1 1 3 3 1 1

Table 7. Field content and multiplicities for the discussed models in this paper.

by a subscript that indicates the symmetry which is broken at this stage. Nevertheless,

these are of course related to the vevs discussed before, but this relation depends on the

class one is discussing. We show this relation explicitly in table 6.

C Model naming scheme

The global naming convention is laid out at the end of section 3. For all configurations

of type g we use a numerical naming scheme. The numbers follow an internal numbering

given by the structure of our Mathematica file. This file is available by the authors upon

request.

Table 7 displays the connection between the multiplicities of the different fields present

below the Planck scale and the model names used in this paper.

– 37 –

Page 39: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

D Specific models

Among the models that are consistent with unification, there is a subset where further in-

teresting conditions are met simultaneously. In this appendix, we list all models within our

framework that (i) show complete GCU at the Planck scale, taken as MPlanck = 1018.2 GeV;

(ii) predict new “exotic” particles at accessible energies Mlow ∼ 10 TeV; and (iii) exhibit a

right-handed neutrino scale in the range 1012 GeV < MNR< 1014 GeV.

Class: F (supersymmetric)

Model #h #F #Φ #Σ #E #T MLR [GeV] MPS [GeV]

F41 0 1 1 1 1 1 ∼ 9 × 1011 ∼ 6 × 1014

F186 1 3 1 1 1 3 ∼ 7 × 1012 ∼ 8 × 1014

F262 3 1 1 3 0 0 ∼ 1 × 1011 ∼ 3 × 1014

Class: B (supersymmetric)

Model #h #F #Φ #Σ #E #T MMSSM [GeV] MLR [GeV] MPS [GeV]

B19 0 0 3 3 0 1 3 × 1010–3 × 1013 31 × 1013–3 × 1015 2 × 1015–8 × 1015

B115 1 1 3 3 0 1 1 × 1011–3 × 1013 3 × 1013–3 × 1015 2 × 1015–6 × 1015

B199 3 3 1 3 0 1 7 × 1011–1 × 1012 2 × 1013–2 × 1014 3 × 1015–4 × 1015

Class: C (supersymmetric)

Model #h #F #Φ #Σ #E #T MMSSM [GeV] MPS [GeV] MQL [GeV]

C43 1 1 3 3 0 1 4 × 1010–3 × 1012 7 × 1015–3 × 1016 7 × 1015–7 × 1017

C53 1 3 1 1 3 1 2 × 1012–1 × 1014 3 × 1016–6 × 1016 1 × 1017–2 × 1018

C127 3 3 1 3 0 1 4 × 1011–4 × 1012 5 × 1015–8 × 1015 5 × 1015–5 × 1017

Non-supersymmetric

Model #h #F #Φ #Σ #E #T MSM [GeV] MPS [GeV] MQL [GeV]

E46 0 1 1 3 0 0 4 × 1012 3 × 1014 −E73 0 3 1 1 0 0 9 × 1010 6 × 1014 −E87 0 3 1 3 1 3 9 × 1010 6 × 1014 −E89 0 3 1 3 3 1 9 × 1010 6 × 1014 −C193 1 3 1 3 0 1 6 × 1010–3 × 1013 2 × 1013–1 × 1014 3 × 1016–4 × 1017

Acknowledgments

This work has been supported by a Coordinated Project Grant of the Faculty of Science

and Technology, University of Siegen. We thank T. Feldmann, C. Luhn and J. Reuter for

discussions and valuable comments on the manuscript.

– 38 –

Page 40: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

Open Access. This article is distributed under the terms of the Creative Commons

Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in

any medium, provided the original author(s) and source are credited.

References

[1] R.N. Mohapatra and J.C. Pati, A natural left-right symmetry, Phys. Rev. D 11 (1975) 2558

[INSPIRE].

[2] G. Senjanovic and R.N. Mohapatra, Exact left-right symmetry and spontaneous violation of

parity, Phys. Rev. D 12 (1975) 1502 [INSPIRE].

[3] J.C. Pati and A. Salam, Lepton number as the fourth color, Phys. Rev. D 10 (1974) 275

[Erratum ibid. D 11 (1975) 703] [INSPIRE].

[4] J.C. Pati and A. Salam, Unified lepton-hadron symmetry and a gauge theory of the basic

interactions, Phys. Rev. D 8 (1973) 1240 [INSPIRE].

[5] H. Georgi and S.L. Glashow, Unity of all elementary particle forces,

Phys. Rev. Lett. 32 (1974) 438 [INSPIRE].

[6] H. Fritzsch and P. Minkowski, Unified interactions of leptons and hadrons,

Annals Phys. 93 (1975) 193 [INSPIRE].

[7] F. Gursey, P. Ramond and P. Sikivie, A universal gauge theory model based on E6,

Phys. Lett. B 60 (1976) 177 [INSPIRE].

[8] M. Malinsky, J.C. Romao and J.W.F. Valle, Supersymmetric SO(10) seesaw mechanism with

low B − L scale, Phys. Rev. Lett. 95 (2005) 161801 [hep-ph/0506296] [INSPIRE].

[9] C. Arbelaez, R.M. Fonseca, M. Hirsch and J.C. Romao, Supersymmetric SO(10)-inspired

GUTs with sliding scales, Phys. Rev. D 87 (2013) 075010 [arXiv:1301.6085] [INSPIRE].

[10] V. De Romeri, M. Hirsch and M. Malinsky, Soft masses in SUSY SO(10) GUTs with low

intermediate scales, Phys. Rev. D 84 (2011) 053012 [arXiv:1107.3412] [INSPIRE].

[11] L. Calibbi, M. Frigerio, S. Lavignac and A. Romanino, Flavour violation in supersymmetric

SO(10) unification with a type-II seesaw mechanism, JHEP 12 (2009) 057

[arXiv:0910.0377] [INSPIRE].

[12] L. Calibbi, L. Ferretti, A. Romanino and R. Ziegler, Gauge coupling unification, the GUT

scale and magic fields, Phys. Lett. B 672 (2009) 152 [arXiv:0812.0342] [INSPIRE].

[13] C. Biggio and L. Calibbi, Phenomenology of SUSY SU(5) with type-I+III seesaw,

JHEP 10 (2010) 037 [arXiv:1007.3750] [INSPIRE].

[14] C. Biggio, L. Calibbi, A. Masiero and S.K. Vempati, Postcards from oases in the desert:

phenomenology of SUSY with intermediate scales, JHEP 08 (2012) 150 [arXiv:1205.6817]

[INSPIRE].

[15] M.R. Buckley and H. Murayama, How can we test seesaw experimentally?,

Phys. Rev. Lett. 97 (2006) 231801 [hep-ph/0606088] [INSPIRE].

[16] J.N. Esteves, S. Kaneko, J.C. Romao, M. Hirsch and W. Porod, Dark matter in minimal

supergravity with type-II seesaw, Phys. Rev. D 80 (2009) 095003 [arXiv:0907.5090]

[INSPIRE].

– 39 –

Page 41: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

[17] J.N. Esteves, J.C. Romao, M. Hirsch, F. Staub and W. Porod, Supersymmetric type-III

seesaw: lepton flavour violating decays and dark matter, Phys. Rev. D 83 (2011) 013003

[arXiv:1010.6000] [INSPIRE].

[18] J.N. Esteves et al., Dark matter and LHC phenomenology in a left-right supersymmetric

model, JHEP 01 (2012) 095 [arXiv:1109.6478] [INSPIRE].

[19] S.K. Majee, M.K. Parida, A. Raychaudhuri and U. Sarkar, Low intermediate scales for

leptogenesis in SUSY SO(10) GUTs, Phys. Rev. D 75 (2007) 075003 [hep-ph/0701109]

[INSPIRE].

[20] F. Braam, J. Reuter and D. Wiesler, The Pati-Salam supersymmetric SM (PSSSM),

AIP Conf. Proc. 1200 (2010) 458 [arXiv:0909.3081] [INSPIRE].

[21] W. Kilian and J. Reuter, Unification without doublet-triplet splitting,

Phys. Lett. B 642 (2006) 81 [hep-ph/0606277] [INSPIRE].

[22] R. Howl and S.F. King, Planck scale unification in a supersymmetric standard model,

Phys. Lett. B 652 (2007) 331 [arXiv:0705.0301] [INSPIRE].

[23] P.S. Bhupal Dev and R.N. Mohapatra, TeV scale inverse seesaw in SO(10) and leptonic

non-unitarity effects, Phys. Rev. D 81 (2010) 013001 [arXiv:0910.3924] [INSPIRE].

[24] M. Lindner and M. Weiser, Gauge coupling unification in left-right symmetric models,

Phys. Lett. B 383 (1996) 405 [hep-ph/9605353] [INSPIRE].

[25] C. Arbelaez, M. Hirsch, M. Malinsky and J.C. Romao, LHC-scale left-right symmetry and

unification, Phys. Rev. D 89 (2014) 035002 [arXiv:1311.3228] [INSPIRE].

[26] F. Siringo, Grand unification in the minimal left-right symmetric extension of the standard

model, Phys. Part. Nucl. Lett. 10 (2013) 94 [arXiv:1208.3599] [INSPIRE].

[27] C.S. Aulakh, B. Bajc, A. Melfo, A. Rasin and G. Senjanovic, Intermediate scales in

supersymmetric GUTs: the survival of the fittest, Phys. Lett. B 460 (1999) 325

[hep-ph/9904352] [INSPIRE].

[28] C.S. Aulakh, B. Bajc, A. Melfo, A. Rasin and G. Senjanovic, SO(10) theory of R-parity and

neutrino mass, Nucl. Phys. B 597 (2001) 89 [hep-ph/0004031] [INSPIRE].

[29] Particle Data Group collaboration, J. Beringer et al., Review of particle physics,

Phys. Rev. D 86 (2012) 010001 [INSPIRE].

[30] S. Weinberg, Baryon and lepton nonconserving processes, Phys. Rev. Lett. 43 (1979) 1566

[INSPIRE].

[31] P. Minkowski, µ→ eγ at a rate of one out of 1-billion muon decays?,

Phys. Lett. B 67 (1977) 421 [INSPIRE].

[32] H. Georgi, Towards a grand unified theory of flavor, Nucl. Phys. B 156 (1979) 126 [INSPIRE].

[33] F. del Aguila and L.E. Ibanez, Higgs bosons in SO(10) and partial unification,

Nucl. Phys. B 177 (1981) 60 [INSPIRE].

[34] R.N. Mohapatra and G. Senjanovic, Higgs boson effects in grand unified theories,

Phys. Rev. D 27 (1983) 1601 [INSPIRE].

[35] S. Dimopoulos and H.M. Georgi, Extended survival hypothesis and fermion masses,

Phys. Lett. B 140 (1984) 67 [INSPIRE].

– 40 –

Page 42: JHEP05(2014)064 - Springer2014)064.pdf · JHEP05(2014)064 Published for ... quark-lepton symmetries that are relevant ... or geometrical structure are required to explain the breaking

JHEP05(2014)064

[36] C.S. Aulakh and R.N. Mohapatra, Implications of supersymmetric SO(10) grand unification,

Phys. Rev. D 28 (1983) 217 [INSPIRE].

[37] R. Slansky, Group theory for unified model building, Phys. Rept. 79 (1981) 1 [INSPIRE].

[38] H.P. Nilles, Supersymmetry, supergravity and particle physics, Phys. Rept. 110 (1984) 1

[INSPIRE].

[39] H.E. Haber and G.L. Kane, The search for supersymmetry: probing physics beyond the

standard model, Phys. Rept. 117 (1985) 75 [INSPIRE].

[40] J. Wess and B. Zumino, A Lagrangian model invariant under supergauge transformations,

Phys. Lett. B 49 (1974) 52 [INSPIRE].

[41] M.T. Grisaru, W. Siegel and M. Rocek, Improved methods for supergraphs,

Nucl. Phys. B 159 (1979) 429 [INSPIRE].

[42] C.S. Aulakh, K. Benakli and G. Senjanovic, Reconciling supersymmetry and left-right

symmetry, Phys. Rev. Lett. 79 (1997) 2188 [hep-ph/9703434] [INSPIRE].

[43] C.S. Aulakh, A. Melfo and G. Senjanovic, Minimal supersymmetric left-right model,

Phys. Rev. D 57 (1998) 4174 [hep-ph/9707256] [INSPIRE].

[44] J.R. Ellis, J.F. Gunion, H.E. Haber, L. Roszkowski and F. Zwirner, Higgs bosons in a

nonminimal supersymmetric model, Phys. Rev. D 39 (1989) 844 [INSPIRE].

[45] H.P. Nilles, M. Srednicki and D. Wyler, Weak interaction breakdown induced by supergravity,

Phys. Lett. B 120 (1983) 346 [INSPIRE].

[46] R.N. Mohapatra and G. Senjanovic, Neutrino mass and spontaneous parity violation,

Phys. Rev. Lett. 44 (1980) 912 [INSPIRE].

[47] M. Gell-Mann, P. Ramond and R. Slansky, Complex spinors and unified theories,

Conf. Proc. C 790927 (1979) 315 [arXiv:1306.4669] [INSPIRE].

[48] D.R.T. Jones, Two-loop β function for a G1 ×G2 gauge theory, Phys. Rev. D 25 (1982) 581

[INSPIRE].

– 41 –